The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of...
Transcript of The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of...
![Page 1: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/1.jpg)
The PA = LU FactorizationMath 218
Brian D. Fitzpatrick
Duke University
November 1, 2019
MATH
![Page 2: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/2.jpg)
Overview
Background“Big Picture” Overview of PA = LUComputational Limitations of rrefRounding ErrorsPermutation MatricesForward Elimination
The PA = LU AlgorithmDescriptionExample
Row EquivalencyA and U are Row-Equivalent
![Page 3: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/3.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 4: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/4.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 5: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/5.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 6: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/6.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 7: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/7.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 8: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/8.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 9: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/9.jpg)
Background“Big Picture” Overview of PA = LU
The PA = LU algorithm produces a factorization used in numericallinear algebra.
I P is a permutation matrix
I L is lower triangular
I U is upper triangular
I PA = LU algorithm is faster than EA = rref(A)
I PA = LU algorithm uses only row swaps and row addition
I row swaps are determined by the method of partial pivoting
I A #»x =#»
b is equivalent to U #»x = #»y where #»y solves L #»y = P#»
b
![Page 10: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/10.jpg)
BackgroundComputational Limitations of rref
RecallThe Gauß-Jordan algorithm produces a matrix factorizationEA = rref(A).
The matrix E is of the form
E = Er · · ·E2E1
where E1,E2, . . . ,Er are the elementary matrices corresponding tothe row operations used in the Gauß-Jordan algorithm.
Utility
Solving A #»x =#»
b is equivalent to solving rref(A) #»x = E#»
b .
![Page 11: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/11.jpg)
BackgroundComputational Limitations of rref
RecallThe Gauß-Jordan algorithm produces a matrix factorizationEA = rref(A). The matrix E is of the form
E = Er · · ·E2E1
where E1,E2, . . . ,Er are the elementary matrices corresponding tothe row operations used in the Gauß-Jordan algorithm.
Utility
Solving A #»x =#»
b is equivalent to solving rref(A) #»x = E#»
b .
![Page 12: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/12.jpg)
BackgroundComputational Limitations of rref
RecallThe Gauß-Jordan algorithm produces a matrix factorizationEA = rref(A). The matrix E is of the form
E = Er · · ·E2E1
where E1,E2, . . . ,Er are the elementary matrices corresponding tothe row operations used in the Gauß-Jordan algorithm.
Utility
Solving A #»x =#»
b is equivalent to solving rref(A) #»x = E#»
b .
![Page 13: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/13.jpg)
BackgroundComputational Limitations of rref
ProblemThe matrix E is the product of a lot of elementary matrices. Themore row operations used, the more time it takes to compute E .
![Page 14: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/14.jpg)
BackgroundComputational Limitations of rref
ExampleThe Gauß-Jordan algorithm gives
9 38 −204 17 −93 17 −11
A
(1/9)·R1→R1−−−−−−−−→
1 38/9 −20/94 17 −93 17 −11
R2 + (−4) · R1 → R2R3 + (−3) · R1 → R3−−−−−−−−−−−−−−−−→
1 38/9 −20/90 1/9 −1/90 13/3 −13/3
9·R2→R2−−−−−−→
1 38/9 −20/90 1 −10 13/3 −13/3
R1 + (−38/9) · R2 → R1R3 + (−13/3) · R2 → R3−−−−−−−−−−−−−−−−−→
1 0 20 1 −10 0 0
rref(A)
The EA = rref(A) factorization is
17 −38 0−4 9 017 −39 1
E
9 38 −204 17 −93 17 −11
A
=
1 0 20 1 −10 0 0
rref(A)
Where did the entries in E come from?
![Page 15: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/15.jpg)
BackgroundComputational Limitations of rref
ExampleThe Gauß-Jordan algorithm gives
9 38 −204 17 −93 17 −11
A
(1/9)·R1→R1−−−−−−−−→
1 38/9 −20/94 17 −93 17 −11
R2 + (−4) · R1 → R2R3 + (−3) · R1 → R3−−−−−−−−−−−−−−−−→
1 38/9 −20/90 1/9 −1/90 13/3 −13/3
9·R2→R2−−−−−−→
1 38/9 −20/90 1 −10 13/3 −13/3
R1 + (−38/9) · R2 → R1R3 + (−13/3) · R2 → R3−−−−−−−−−−−−−−−−−→
1 0 20 1 −10 0 0
rref(A)
The EA = rref(A) factorization is
17 −38 0−4 9 017 −39 1
E
9 38 −204 17 −93 17 −11
A
=
1 0 20 1 −10 0 0
rref(A)
Where did the entries in E come from?
![Page 16: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/16.jpg)
BackgroundComputational Limitations of rref
ExampleThe Gauß-Jordan algorithm gives
9 38 −204 17 −93 17 −11
A
(1/9)·R1→R1−−−−−−−−→
1 38/9 −20/94 17 −93 17 −11
R2 + (−4) · R1 → R2R3 + (−3) · R1 → R3−−−−−−−−−−−−−−−−→
1 38/9 −20/90 1/9 −1/90 13/3 −13/3
9·R2→R2−−−−−−→
1 38/9 −20/90 1 −10 13/3 −13/3
R1 + (−38/9) · R2 → R1R3 + (−13/3) · R2 → R3−−−−−−−−−−−−−−−−−→
1 0 20 1 −10 0 0
rref(A)
The EA = rref(A) factorization is
17 −38 0−4 9 017 −39 1
E
9 38 −204 17 −93 17 −11
A
=
1 0 20 1 −10 0 0
rref(A)
Where did the entries in E come from?
![Page 17: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/17.jpg)
BackgroundRounding Errors
NoteUsing rational numbers to reduce a system produces an exactanswer.
rref
7/1000 306/5 93/1000 613/10481/100 −148/25 111/100 0
407/5 28/25 59/50 837/10
=
1 0 0 10 1 0 10 0 1 1
![Page 18: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/18.jpg)
BackgroundRounding Errors
In practice, computers use floating point numbers to store data.After each computation, the computer rounds each entry to agiven number of significant digits.
![Page 19: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/19.jpg)
BackgroundRounding Errors
Applying Gauß-Jordan algorithm to the previous example withrounding to three significant digits gives
0.007 61.2 0.093 61.34.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→ 1.0 8740.0 13.3 8760.0
4.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→
1.0 8740.0 13.3 8760.00.0 −42000.0 −62.9 −42100.00.0 −711000.0 −1080.0 −713000.0
→
1.0 8740.0 13.3 8760.00.0 1.0 0.0015 1.00.0 −711000.0 −1080.0 −713000.0
→
1.0 0.0 0.19 20.00.0 1.0 0.0015 1.00.0 0.0 −13.5 −2000.0
→
1.0 0.0 0.19 20.00.0 1.0 0.0015 1.00.0 0.0 1.0 148.0
→
1.0 0.0 0.0 −8.120.0 1.0 0.0 0.7780.0 0.0 1.0 148.0
The exact solution is #»x = 〈1, 1, 1〉 . Rounding “destabilized” thealgorithm.
![Page 20: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/20.jpg)
BackgroundRounding Errors
Applying Gauß-Jordan algorithm to the previous example withrounding to three significant digits gives
0.007 61.2 0.093 61.34.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→ 1.0 8740.0 13.3 8760.0
4.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→
1.0 8740.0 13.3 8760.00.0 −42000.0 −62.9 −42100.00.0 −711000.0 −1080.0 −713000.0
→
1.0 8740.0 13.3 8760.00.0 1.0 0.0015 1.00.0 −711000.0 −1080.0 −713000.0
→
1.0 0.0 0.19 20.00.0 1.0 0.0015 1.00.0 0.0 −13.5 −2000.0
→
1.0 0.0 0.19 20.00.0 1.0 0.0015 1.00.0 0.0 1.0 148.0
→
1.0 0.0 0.0 −8.120.0 1.0 0.0 0.7780.0 0.0 1.0 148.0
The exact solution is #»x = 〈1, 1, 1〉 .
Rounding “destabilized” thealgorithm.
![Page 21: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/21.jpg)
BackgroundRounding Errors
Applying Gauß-Jordan algorithm to the previous example withrounding to three significant digits gives
0.007 61.2 0.093 61.34.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→ 1.0 8740.0 13.3 8760.0
4.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→
1.0 8740.0 13.3 8760.00.0 −42000.0 −62.9 −42100.00.0 −711000.0 −1080.0 −713000.0
→
1.0 8740.0 13.3 8760.00.0 1.0 0.0015 1.00.0 −711000.0 −1080.0 −713000.0
→
1.0 0.0 0.19 20.00.0 1.0 0.0015 1.00.0 0.0 −13.5 −2000.0
→
1.0 0.0 0.19 20.00.0 1.0 0.0015 1.00.0 0.0 1.0 148.0
→
1.0 0.0 0.0 −8.120.0 1.0 0.0 0.7780.0 0.0 1.0 148.0
The exact solution is #»x = 〈1, 1, 1〉 . Rounding “destabilized” thealgorithm.
![Page 22: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/22.jpg)
BackgroundRounding Errors
The method of partial pivoting uses the largest possible number (inabsolute value) to create pivots. This minimizes rounding errors.
![Page 23: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/23.jpg)
BackgroundRounding Errors
Using partial pivoting, we have 0.007 61.2 0.093 61.34.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→ 81.4 1.12 1.18 83.7
4.81 −5.92 1.11 0.00.007 61.2 0.093 61.3
→
1.0 0.0138 0.0145 1.034.81 −5.92 1.11 0.0
0.007 61.2 0.093 61.3
→ 1.0 0.0138 0.0145 1.03
0.0 −5.99 1.04 −4.950.0 61.2 0.0929 61.3
→
1.0 0.0138 0.0145 1.030.0 61.2 0.0929 61.30.0 −5.99 1.04 −4.95
→ 1.0 0.0138 0.0145 1.03
0.0 1.0 0.00152 1.00.0 −5.99 1.04 −4.95
→
1.0 0.0 0.0145 1.020.0 1.0 0.00152 1.00.0 0.0 1.05 1.04
→ 1.0 0.0 0.0145 1.02
0.0 1.0 0.00152 1.00.0 0.0 1.0 0.99
→
1.0 0.0 0.0 1.010.0 1.0 0.0 0.9980.0 0.0 1.0 0.99
By using partial pivoting, we have reduced the rounding error.
![Page 24: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/24.jpg)
BackgroundRounding Errors
Using partial pivoting, we have 0.007 61.2 0.093 61.34.81 −5.92 1.11 0.081.4 1.12 1.18 83.7
→ 81.4 1.12 1.18 83.7
4.81 −5.92 1.11 0.00.007 61.2 0.093 61.3
→
1.0 0.0138 0.0145 1.034.81 −5.92 1.11 0.0
0.007 61.2 0.093 61.3
→ 1.0 0.0138 0.0145 1.03
0.0 −5.99 1.04 −4.950.0 61.2 0.0929 61.3
→
1.0 0.0138 0.0145 1.030.0 61.2 0.0929 61.30.0 −5.99 1.04 −4.95
→ 1.0 0.0138 0.0145 1.03
0.0 1.0 0.00152 1.00.0 −5.99 1.04 −4.95
→
1.0 0.0 0.0145 1.020.0 1.0 0.00152 1.00.0 0.0 1.05 1.04
→ 1.0 0.0 0.0145 1.02
0.0 1.0 0.00152 1.00.0 0.0 1.0 0.99
→
1.0 0.0 0.0 1.010.0 1.0 0.0 0.9980.0 0.0 1.0 0.99
By using partial pivoting, we have reduced the rounding error.
![Page 25: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/25.jpg)
BackgroundPermutation Matrices
DefinitionA permutation matrix is a matrix obtained by performing rowswaps on an identity matrix.
Example
[1 00 1
] 0 0 10 1 01 0 0
0 1 0 00 0 0 10 0 1 01 0 0 0
![Page 26: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/26.jpg)
BackgroundPermutation Matrices
DefinitionA permutation matrix is a matrix obtained by performing rowswaps on an identity matrix.
Example
[1 00 1
] 0 0 10 1 01 0 0
0 1 0 00 0 0 10 0 1 01 0 0 0
![Page 27: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/27.jpg)
BackgroundPermutation Matrices
TheoremLet P be an elementary matrix corresponding to a row swap. ThenP is a permutation matrix, P is symmetric (Pᵀ = P), and P is selfinverse (P2 = I ).
![Page 28: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/28.jpg)
BackgroundPermutation Matrices
TheoremLet P be a permutation matrix. Then P is the product ofelementary matrices corresponding to row swaps.
![Page 29: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/29.jpg)
BackgroundPermutation Matrices
DefinitionAn orthogonal matrix is a matrix Q satisfying Q−1 = Qᵀ.
Example
Consider the matrix Q given by
Q =
[35/37 12/37−12/37 35/37
]Then Q is orthogonal since
QQᵀ =
[35/37 12/37−12/37 35/37
] [35/37 −12/3712/37 35/37
]=
[1 00 1
]
![Page 30: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/30.jpg)
BackgroundPermutation Matrices
DefinitionAn orthogonal matrix is a matrix Q satisfying Q−1 = Qᵀ.
Example
Consider the matrix Q given by
Q =
[35/37 12/37−12/37 35/37
]
Then Q is orthogonal since
QQᵀ =
[35/37 12/37−12/37 35/37
] [35/37 −12/3712/37 35/37
]=
[1 00 1
]
![Page 31: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/31.jpg)
BackgroundPermutation Matrices
DefinitionAn orthogonal matrix is a matrix Q satisfying Q−1 = Qᵀ.
Example
Consider the matrix Q given by
Q =
[35/37 12/37−12/37 35/37
]Then Q is orthogonal since
QQᵀ =
[35/37 12/37−12/37 35/37
] [35/37 −12/3712/37 35/37
]=
[1 00 1
]
![Page 32: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/32.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk .
Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 33: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/33.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk .
Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 34: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/34.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible.
Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 35: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/35.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 36: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/36.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 =
(P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 37: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/37.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
=
P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 38: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/38.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
=
Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 39: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/39.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
=
Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 40: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/40.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
=
(P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 41: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/41.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
=
Pᵀ
![Page 42: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/42.jpg)
BackgroundPermutation Matrices
TheoremEvery permutation matrix is an orthogonal matrix.
Proof.Write P as the product of elementary matrices corresponding torow swaps P = P1P2 · · ·Pk . Then P is invertible since each Pi isinvertible. Furthermore, we have
P−1 = (P1P2 · · ·Pk)−1
= P−1k · · ·P−12 P−11
= Pk · · ·P2P1
= Pᵀk · · ·P
ᵀ2P
ᵀ1
= (P1P2 · · ·Pk)ᵀ
= Pᵀ
![Page 43: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/43.jpg)
BackgroundForward Elimination
The PA = LU algorithm uses forward elimination.
![Page 44: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/44.jpg)
BackgroundForward Elimination
Consider the row reductions 1 −2 11 0 2−1 1 0
A
R2 − R1 → R2R3 + R1 → R3−−−−−−−−−→
1 −2 10 2 10 −1 1
R3+(1/2)·R2→R3−−−−−−−−−−→
1 −2 10 2 10 0 3/2
U
The EA = U factorization is 1 0 0−1 1 01/2 1/2 1
E
1 −2 11 0 2−1 1 0
A
=
1 −2 10 2 10 0 3/2
U
Putting L = E−1 gives A = LU where 1 −2 11 0 2−1 1 0
A
=
1 0 01 1 0−1 −1/2 1
L
1 −2 10 2 10 0 3/2
U
The entries of L are the “multipliers” used in the row reductions.
![Page 45: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/45.jpg)
BackgroundForward Elimination
Consider the row reductions 1 −2 11 0 2−1 1 0
A
R2 − R1 → R2R3 + R1 → R3−−−−−−−−−→
1 −2 10 2 10 −1 1
R3+(1/2)·R2→R3−−−−−−−−−−→
1 −2 10 2 10 0 3/2
U
The EA = U factorization is 1 0 0−1 1 01/2 1/2 1
E
1 −2 11 0 2−1 1 0
A
=
1 −2 10 2 10 0 3/2
U
Putting L = E−1 gives A = LU where 1 −2 11 0 2−1 1 0
A
=
1 0 01 1 0−1 −1/2 1
L
1 −2 10 2 10 0 3/2
U
The entries of L are the “multipliers” used in the row reductions.
![Page 46: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/46.jpg)
BackgroundForward Elimination
Consider the row reductions 1 −2 11 0 2−1 1 0
A
R2 − R1 → R2R3 + R1 → R3−−−−−−−−−→
1 −2 10 2 10 −1 1
R3+(1/2)·R2→R3−−−−−−−−−−→
1 −2 10 2 10 0 3/2
U
The EA = U factorization is 1 0 0−1 1 01/2 1/2 1
E
1 −2 11 0 2−1 1 0
A
=
1 −2 10 2 10 0 3/2
U
Putting L = E−1 gives A = LU where 1 −2 11 0 2−1 1 0
A
=
1 0 01 1 0−1 −1/2 1
L
1 −2 10 2 10 0 3/2
U
The entries of L are the “multipliers” used in the row reductions.
![Page 47: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/47.jpg)
BackgroundForward Elimination
Consider the row reductions 1 −2 11 0 2−1 1 0
A
R2 − R1 → R2R3 + R1 → R3−−−−−−−−−→
1 −2 10 2 10 −1 1
R3+(1/2)·R2→R3−−−−−−−−−−→
1 −2 10 2 10 0 3/2
U
The EA = U factorization is 1 0 0−1 1 01/2 1/2 1
E
1 −2 11 0 2−1 1 0
A
=
1 −2 10 2 10 0 3/2
U
Putting L = E−1 gives A = LU where 1 −2 11 0 2−1 1 0
A
=
1 0 01 1 0−1 −1/2 1
L
1 −2 10 2 10 0 3/2
U
The entries of L are the “multipliers” used in the row reductions.
![Page 48: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/48.jpg)
The PA = LU AlgorithmDescription
Algorithm (PA = LU Factorization with Partial Pivoting)
Suppose A is m × n. Start with i = j = 1 and L = P = Im.
Step 1 Find k ≥ i where |akj | > 0 is as large as possible.Perform Ri ↔ Rk on A, P, and the multipliers in L.If not possible, increase j by one and repeat this step.
Step 2 Eliminate all entries below aij by subtracting suitiblemultiples mkj of Rowi . Put `kj = −mkj .
Step 3 Increase i and j by one and return to Step 1.
The algorithm terminates after the last row or column is processed.
![Page 49: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/49.jpg)
The PA = LU AlgorithmDescription
Algorithm (PA = LU Factorization with Partial Pivoting)
Suppose A is m × n. Start with i = j = 1 and L = P = Im.
Step 1 Find k ≥ i where |akj | > 0 is as large as possible.Perform Ri ↔ Rk on A, P, and the multipliers in L.If not possible, increase j by one and repeat this step.
Step 2 Eliminate all entries below aij by subtracting suitiblemultiples mkj of Rowi . Put `kj = −mkj .
Step 3 Increase i and j by one and return to Step 1.
The algorithm terminates after the last row or column is processed.
![Page 50: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/50.jpg)
The PA = LU AlgorithmDescription
Algorithm (PA = LU Factorization with Partial Pivoting)
Suppose A is m × n. Start with i = j = 1 and L = P = Im.
Step 1 Find k ≥ i where |akj | > 0 is as large as possible.Perform Ri ↔ Rk on A, P, and the multipliers in L.If not possible, increase j by one and repeat this step.
Step 2 Eliminate all entries below aij by subtracting suitiblemultiples mkj of Rowi . Put `kj = −mkj .
Step 3 Increase i and j by one and return to Step 1.
The algorithm terminates after the last row or column is processed.
![Page 51: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/51.jpg)
The PA = LU AlgorithmDescription
Algorithm (PA = LU Factorization with Partial Pivoting)
Suppose A is m × n. Start with i = j = 1 and L = P = Im.
Step 1 Find k ≥ i where |akj | > 0 is as large as possible.Perform Ri ↔ Rk on A, P, and the multipliers in L.If not possible, increase j by one and repeat this step.
Step 2 Eliminate all entries below aij by subtracting suitiblemultiples mkj of Rowi . Put `kj = −mkj .
Step 3 Increase i and j by one and return to Step 1.
The algorithm terminates after the last row or column is processed.
![Page 52: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/52.jpg)
The PA = LU AlgorithmDescription
QuestionHow does PA = LU help us solve A #»x =
#»
b ?
AnswerA #»x =
#»
b is equivalent to PA #»x = P#»
b . This gives LU #»x = P#»
b .
Step 1 Use “forward substitution” to solve L #»y = P#»
b for #»y .
Step 2 Use “back substitution” to solve U #»x = #»y for #»x .
![Page 53: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/53.jpg)
The PA = LU AlgorithmDescription
QuestionHow does PA = LU help us solve A #»x =
#»
b ?
AnswerA #»x =
#»
b is equivalent to PA #»x = P#»
b . This gives LU #»x = P#»
b .
Step 1 Use “forward substitution” to solve L #»y = P#»
b for #»y .
Step 2 Use “back substitution” to solve U #»x = #»y for #»x .
![Page 54: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/54.jpg)
The PA = LU AlgorithmDescription
QuestionHow does PA = LU help us solve A #»x =
#»
b ?
AnswerA #»x =
#»
b is equivalent to PA #»x = P#»
b . This gives LU #»x = P#»
b .
Step 1 Use “forward substitution” to solve L #»y = P#»
b for #»y .
Step 2 Use “back substitution” to solve U #»x = #»y for #»x .
![Page 55: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/55.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2
−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 56: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/56.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2
−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 57: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/57.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 58: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/58.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 00 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 59: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/59.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 01 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 60: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/60.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 61: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/61.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3
−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 62: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/62.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 63: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/63.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 64: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/64.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 65: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/65.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 66: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/66.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3
−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 67: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/67.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 68: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/68.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 69: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/69.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 70: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/70.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 71: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/71.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3
−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 72: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/72.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 73: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/73.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 74: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/74.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 75: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/75.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 76: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/76.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 77: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/77.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 78: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/78.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 79: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/79.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 80: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/80.jpg)
The PA = LU AlgorithmExample
The PA = LU agorithm with partial pivoting gives
A 1 2 04 −3 −53 2 13
R1↔R2−−−−−→
U 4 −3 −51 2 03 2 13
L 1 0 0
0 1 00 0 1
P 0 1 0
1 0 00 0 1
R2 − (1/4) · R1 → R2R3 − (3/4) · R1 → R3−−−−−−−−−−−−−−−→
4 −3 −50 11/4 5/40 17/4 67/4
1 0 01/4 1 03/4 0 1
0 1 01 0 00 0 1
R2↔R3−−−−−→
4 −3 −50 17/4 67/40 11/4 5/4
1 0 03/4 1 01/4 0 1
0 1 00 0 11 0 0
R3−(11/17)·R2→R3−−−−−−−−−−−−→
4 −3 −50 17/4 67/40 0 −163/17
1 0 03/4 1 01/4 11/17 1
0 1 00 0 11 0 0
This gives the PA = LU factorization
0 1 00 0 11 0 0
P
1 2 04 −3 −53 2 13
A
=
1 0 03/4 1 01/4 11/17 1
L
4 −3 −50 17/4 67/40 0 −163/17
U
![Page 81: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/81.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 82: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/82.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326
→ y1 = −326
(3/4) · y1 + y2 = −163
→ y2 = 163/2
(1/4) · y1 + (11/17) · y2 + y3 = 326
→ y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 83: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/83.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163
→ y2 = 163/2
(1/4) · y1 + (11/17) · y2 + y3 = 326
→ y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 84: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/84.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326
→ y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 85: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/85.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 86: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/86.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 87: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/87.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326
→ x1 = −4
(17/4) · x2 + (67/4) · x3 = 163/2
→ x2 = 165
(−163/17) · x3 = 6031/17
→ x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 88: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/88.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326
→ x1 = −4
(17/4) · x2 + (67/4) · x3 = 163/2
→ x2 = 165
(−163/17) · x3 = 6031/17 → x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 89: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/89.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326
→ x1 = −4
(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165(−163/17) · x3 = 6031/17 → x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 90: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/90.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17 → x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 91: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/91.jpg)
The PA = LU AlgorithmExample
For#»
b = 〈326, −326, −163〉 , the system L #»y = P#»
b is
y1 = −326 → y1 = −326(3/4) · y1 + y2 = −163 → y2 = 163/2(1/4) · y1 + (11/17) · y2 + y3 = 326 → y3 = 6031/17
For #»y = 〈−326, 163/2, 6031/17〉 , the system U #»x = #»y is
4 x1 − 3 x2 − 5 x3 = −326 → x1 = −4(17/4) · x2 + (67/4) · x3 = 163/2 → x2 = 165
(−163/17) · x3 = 6031/17 → x3 = −37
The solution to A #»x =#»
b is #»x = 〈−4, 165, −37〉 .
![Page 92: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/92.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 93: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/93.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
A
op1−−→ A1op2−−→ A2
op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 94: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/94.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 95: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/95.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2
op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 96: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/96.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · ·
opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 97: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/97.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 98: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/98.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent.
In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 99: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/99.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) =
rank(U) and nullity(A) = nullity(U).
![Page 100: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/100.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U)
and nullity(A) = nullity(U).
![Page 101: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/101.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) =
nullity(U).
![Page 102: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/102.jpg)
Row EquivalencyA and U are Row-Equivalent
NoteThe U in PA = LU factorization is obtained by performingelementary row operations on A.
Aop1−−→ A1
op2−−→ A2op3−−→ · · · opr−−→ U
This means that A and U are row equivalent. In particular,rank(A) = rank(U) and nullity(A) = nullity(U).
![Page 103: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/103.jpg)
Row EquivalencyA and U are Row-Equivalent
TheoremGiven PA = LU, we have
rank(A) = rank(U) nullity(A) = nullity(U)
Advantage
We can compute U faster than we can compute rref(A). ThePA = LU algorithm improves our rank algorithm!
![Page 104: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/104.jpg)
Row EquivalencyA and U are Row-Equivalent
TheoremGiven PA = LU, we have
rank(A) = rank(U) nullity(A) = nullity(U)
Advantage
We can compute U faster than we can compute rref(A).
ThePA = LU algorithm improves our rank algorithm!
![Page 105: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/105.jpg)
Row EquivalencyA and U are Row-Equivalent
TheoremGiven PA = LU, we have
rank(A) = rank(U) nullity(A) = nullity(U)
Advantage
We can compute U faster than we can compute rref(A). ThePA = LU algorithm improves our rank algorithm!
![Page 106: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/106.jpg)
Row EquivalencyA and U are Row-Equivalent
Example
Consider the PA = LU factorization
P0 1 0 00 0 0 11 0 0 00 0 1 0
A
1 −5 −4 −21 −18 −14 −6−2 11 16 68 54 45 −2
0 0 1 3 2 2 −2−2 13 16 72 58 47 2
=
L1 0 0 01 1 0 0
−1/2 1/4 1 00 0 1/4 1
U
−2 11 16 68 54 45 −20 2 0 4 4 2 40 0 4 12 8 8 −80 0 0 0 0 0 0
Here, we have
rank(A) = 3 nullity(A) = 4
![Page 107: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/107.jpg)
Row EquivalencyA and U are Row-Equivalent
Example
Consider the PA = LU factorization
P0 1 0 00 0 0 11 0 0 00 0 1 0
A
1 −5 −4 −21 −18 −14 −6−2 11 16 68 54 45 −2
0 0 1 3 2 2 −2−2 13 16 72 58 47 2
=
L1 0 0 01 1 0 0
−1/2 1/4 1 00 0 1/4 1
U
−2 11 16 68 54 45 −20 2 0 4 4 2 40 0 4 12 8 8 −80 0 0 0 0 0 0
Here, we have
rank(A) = 3 nullity(A) = 4
![Page 108: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/108.jpg)
Row EquivalencyA and U are Row-Equivalent
Example
Consider the PA = LU factorization
P0 1 0 00 0 0 11 0 0 00 0 1 0
A
1 −5 −4 −21 −18 −14 −6−2 11 16 68 54 45 −2
0 0 1 3 2 2 −2−2 13 16 72 58 47 2
=
L1 0 0 01 1 0 0
−1/2 1/4 1 00 0 1/4 1
U
−2 11 16 68 54 45 −20 2 0 4 4 2 40 0 4 12 8 8 −80 0 0 0 0 0 0
Here, we have
rank(A) =
3 nullity(A) = 4
![Page 109: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/109.jpg)
Row EquivalencyA and U are Row-Equivalent
Example
Consider the PA = LU factorization
P0 1 0 00 0 0 11 0 0 00 0 1 0
A
1 −5 −4 −21 −18 −14 −6−2 11 16 68 54 45 −2
0 0 1 3 2 2 −2−2 13 16 72 58 47 2
=
L1 0 0 01 1 0 0
−1/2 1/4 1 00 0 1/4 1
U
−2 11 16 68 54 45 −20 2 0 4 4 2 40 0 4 12 8 8 −80 0 0 0 0 0 0
Here, we have
rank(A) = 3
nullity(A) = 4
![Page 110: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/110.jpg)
Row EquivalencyA and U are Row-Equivalent
Example
Consider the PA = LU factorization
P0 1 0 00 0 0 11 0 0 00 0 1 0
A
1 −5 −4 −21 −18 −14 −6−2 11 16 68 54 45 −2
0 0 1 3 2 2 −2−2 13 16 72 58 47 2
=
L1 0 0 01 1 0 0
−1/2 1/4 1 00 0 1/4 1
U
−2 11 16 68 54 45 −20 2 0 4 4 2 40 0 4 12 8 8 −80 0 0 0 0 0 0
Here, we have
rank(A) = 3 nullity(A) =
4
![Page 111: The PA=LU Factorization - Math 218bfitzpat/teaching/218s20/...Background \Big Picture" Overview of PA = LU The PA = LU algorithm produces a factorization used in numerical linear algebra.](https://reader033.fdocuments.in/reader033/viewer/2022050502/5f94d532b371ca1b125649bf/html5/thumbnails/111.jpg)
Row EquivalencyA and U are Row-Equivalent
Example
Consider the PA = LU factorization
P0 1 0 00 0 0 11 0 0 00 0 1 0
A
1 −5 −4 −21 −18 −14 −6−2 11 16 68 54 45 −2
0 0 1 3 2 2 −2−2 13 16 72 58 47 2
=
L1 0 0 01 1 0 0
−1/2 1/4 1 00 0 1/4 1
U
−2 11 16 68 54 45 −20 2 0 4 4 2 40 0 4 12 8 8 −80 0 0 0 0 0 0
Here, we have
rank(A) = 3 nullity(A) = 4