ECIV 520 Structural Analysis II Review of Matrix Algebra.
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Transcript of ECIV 520 Structural Analysis II Review of Matrix Algebra.
![Page 1: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/1.jpg)
ECIV 520
Structural Analysis II
Review of Matrix Algebra
![Page 2: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/2.jpg)
Linear Equations in Matrix Form
10z8y3x5
6z3yx12
24z23y6x10
![Page 3: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/3.jpg)
Linear Equations in Matrix Form
10z8y3x5
10
z
y
x
835
![Page 4: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/4.jpg)
Linear Equations in Matrix Form
6
z
y
x
3112
6z3yx12
![Page 5: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/5.jpg)
Linear Equations in Matrix Form
24
z
y
x
23610
24z23y6x10
![Page 6: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/6.jpg)
23610
3112
835
z
y
x
24
6
10
10
z
y
x
835
6
z
y
x
3112
24
z
y
x
23610
![Page 7: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/7.jpg)
Matrix Algebra
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
Rectangular Array of Elements Represented by a single symbol [A]
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Matrix Algebra
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
Row 1
Row 3
Column 2 Column m
n x m Matrix
![Page 9: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/9.jpg)
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
Matrix Algebra
32a
3rd Row
2nd Column
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Matrix Algebra
m321 bbbbB
1 Row, m Columns
Row Vector
B
![Page 11: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/11.jpg)
Matrix Algebra
n
3
2
1
c
c
c
c
C
n Rows, 1 Column
Column Vector
C
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Matrix Algebra
5554535251
4544434241
3534333231
2524232221
1514131211
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
A
If n = m Square Matrix
e.g. n=m=5e.g. n=m=5Main Diagonal
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Matrix Algebra
9264
2732
6381
4215
A
Special Types of Square Matrices
Symmetric: aSymmetric: aijij = a = ajiji
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Matrix Algebra
9000
0700
0080
0005
A
Diagonal: aDiagonal: aijij = 0, i = 0, ijj
Special Types of Square Matrices
![Page 15: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/15.jpg)
Matrix Algebra
1000
0100
0010
0001
I
Identity: aIdentity: aiiii=1.0 a=1.0 aijij = 0, i = 0, ijj
Special Types of Square Matrices
![Page 16: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/16.jpg)
nm
m333
m22322
m1131211
a000
aa00
aaa0
aaaa
A
Matrix Algebra
Upper TriangularUpper Triangular
Special Types of Square Matrices
![Page 17: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/17.jpg)
nm3n2n1n
333231
2221
11
aaaa
0aaa
00aa
000a
A
Matrix Algebra
Lower TriangularLower Triangular
Special Types of Square Matrices
![Page 18: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/18.jpg)
nm
3332
232221
1211
a000
0aa0
0aaa
00aa
A
Matrix Algebra
BandedBanded
Special Types of Square Matrices
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Matrix Operating Rules - Equality
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
pq3p2p1p
q3333231
q2232221
q1131211
bbbb
bbab
bbbb
bbbb
B
[A]mxn=[B]pxq
n=p m=q aij=bij
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Matrix Operating Rules - Addition
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
pq3p2p1p
q3333231
q2232221
q1131211
bbbb
bbab
bbbb
bbbb
B
[C]mxn= [A]mxn+[B]pxq
n=p
m=qcij = aij+bij
![Page 21: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/21.jpg)
Matrix Operating Rules - Addition
Properties
[A]+[B] = [B]+[A]
[A]+([B]+[C]) = ([A]+[B])+[C]
![Page 22: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/22.jpg)
Multiplication by Scalar
nm3n2n1n
m3333231
m2232221
m1131211
gagagaga
gagagaga
gagagaga
gagagaga
AgD
![Page 23: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/23.jpg)
Matrix Multiplication
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
pq3p2p1p
q3333231
q2232221
q1131211
bbbb
bbab
bbbb
bbbb
B
[A] n x m . [B] p x q = [C] n x q
m=p
n
1kkjikij bac
![Page 24: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/24.jpg)
Matrix Multiplication
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
pq3p2p1p
q3333231
q2232221
q1131211
bbbb
bbab
bbbb
bbbb
B
1nn13113
2112111111
baba
babac
11c
C
![Page 25: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/25.jpg)
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
Matrix Multiplication
pq3p2p1p
q3333231
q2232221
q1131211
bbbb
bbab
bbbb
bbbb
B
3nn23323
2322132123
baba
babac
23c
C
![Page 26: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/26.jpg)
Matrix Multiplication - Properties
Associative: [A]([B][C]) = ([A][B])[C]
If dimensions suitable
Distributive: [A]([B]+[C]) = [A][B]+[A] [C]
Attention: [A][B] [B][A]
![Page 27: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/27.jpg)
nmm3m2m1
3n332313
2n322212
1n312111
T
aaaa
aaaa
aaaa
aaaa
A
nm3n2n1n
m3333231
m2232221
m1131211
aaaa
aaaa
aaaa
aaaa
A
Operations - Transpose
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Operations - Trace
5554535251
4544434241
3534333231
2524232221
1514131211
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
A
Square Matrix
tr[A] = tr[A] = aaiiii
![Page 29: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/29.jpg)
Determinants
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
A
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
det
AA
Are composed of same elements
Completely Different Mathematical Concept
![Page 30: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/30.jpg)
Determinants
2221
1211
aa
aaA
Defined in a recursive form
2x2 matrix
122122112221
1211det aaaaaa
aaA
![Page 31: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/31.jpg)
Determinants
nnnjnn
inijii
nj
nj
aaaa
aaaa
aaaa
aaaa
21
21
222221
111211
A
ijAMinor
![Page 32: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/32.jpg)
Determinants
ijji
ij AACofactor minor1
n
kikik AaA
1
]det[
![Page 33: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/33.jpg)
DeterminantsDefined in a recursive form
3x3 matrix
3231
222113
3331
232112
3332
232211
det
aa
aaa
aa
aaa
aa
aaa
A
333231
232221
131211
aaa
aaa
aaa
![Page 34: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/34.jpg)
333231
232221
131211
aaa
aaa
aaa
Determinants
3332
232211 aa
aaa
3231
222113
3331
232112 aa
aaa
aa
aaa
3332
2322
aa
aaMinor a11
![Page 35: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/35.jpg)
333231
232221
131211
aaa
aaa
aaa
Determinants
3331
2321
aa
aaMinor a12
3332
232211 aa
aaa
3331
232112 aa
aaa
3231
222113 aa
aaa
![Page 36: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/36.jpg)
333231
232221
131211
aaa
aaa
aaa
Determinants
3231
2221
aa
aaMinor a13
3332
232211 aa
aaa
3331
232112 aa
aaa
3231
222113 aa
aaa
![Page 37: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/37.jpg)
DeterminantsProperties1) If two rows or two columns of matrix [A] are equal then det[A]=0
2) Interchanging any two rows or columns will change the sign of the det
3) If a row or a column of a matrix is {0} then det[A]=0
4)
5) If we multiply any row or column by a scalar s then
6) If any row or column is replaced by a linear combination of any of the other rows or columns the value of det[A] remains unchanged
AsAs n
AsA
![Page 38: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/38.jpg)
Operations - Inverse
[A] [A]-1
[A] [A]-1=[I]
If [A]-1 does not exist[A] is singular
![Page 39: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/39.jpg)
Operations - Inverse
Calculation of [A]-1
AadjA
A11
TijAAadj
![Page 40: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/40.jpg)
Solution of Linear Equations
23610
3112
835
z
y
x
24
6
10
bxA
bAxAA 11
bAxI 1 bAx 1
![Page 41: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/41.jpg)
Numerical Solution of Linear Equations
![Page 42: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/42.jpg)
Solution of Linear Equations
9835 zyx
7310 zyx
10500 zyx
Consider the system
![Page 43: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/43.jpg)
Solution of Linear Equations
9835 zyx
730 zyx
10500 zyx
25
10 z
![Page 44: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/44.jpg)
Solution of Linear Equations
9835 zyx
730 zyx
2z
7230 yx
167 y
![Page 45: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/45.jpg)
Solution of Linear Equations9835 zyx
2z
1y
928135 x
25
1639
x 2x
![Page 46: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/46.jpg)
Solution of Linear Equations
10
7
9
500
310
835
z
y
x
Express In Matrix Form
Upper Triangular
What is the characteristic?
Solution by Back Substitution
![Page 47: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/47.jpg)
Solution of Linear EquationsObjective
Can we express any system of equations in a form
nnnn
n
n
n
b
b
b
b
x
x
x
x
a
aa
aaa
aaaa
3
2
1
3
2
1
333
22322
1131211
000
00
0
0
![Page 48: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/48.jpg)
BackgroundConsider
1035 yx(Eq 1)
5810 yx(Eq 2)
Solution
5.7
5.6
y
x
20610 yx2*(Eq 1)
5810 yx(Eq 2)
Solution
5.7
5.6
y
x!!!!!!
Scaling Does Not Change the SolutionScaling Does Not Change the Solution
![Page 49: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/49.jpg)
BackgroundConsider
20610 yx(Eq 1)
152 y(Eq 2)-(Eq 1)
Solution
5.7
5.6
y
x!!!!!!
20610 yx(Eq 1)
5810 yx(Eq 2)
Solution
5.7
5.6
y
x
Operations Do Not Change the SolutionOperations Do Not Change the Solution
![Page 50: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/50.jpg)
Gauss Elimination
10835 zyx
2423610 zyx
6312 zyx
Example
Forward Elimination
![Page 51: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/51.jpg)
Gauss Elimination
10835 zyx
24z23y6x10
zyx 835
5
1210
5
12
6312 zyx
-
305
81
5
310 zyx 302.162.60 zyx
![Page 52: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/52.jpg)
Gauss Elimination
10835 zyx
24z23y6x10
6312 zyx 302.162.60 zyx
Substitute 2nd eq with new
![Page 53: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/53.jpg)
Gauss Elimination
10835 zyx
24z23y6x10
302.162.60 zyx
zyx 835
5
1010
5
10-
439120 zyx
![Page 54: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/54.jpg)
Gauss Elimination
10835 zyx
24z23y6x10
302.162.60 zyx
Substitute 3rd eq with new
439120 zyx
![Page 55: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/55.jpg)
Gauss Elimination
10835 zyx
302.162.60 zyx
439120 zyx
zy 2.162.6
2.6
12 30
2.6
12-
064.62645.700 zyx
![Page 56: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/56.jpg)
Gauss Elimination
10835 zyx
30970 zyx
Substitute 3rd eq with new
439120 zyx 064.62645.700 zyx
![Page 57: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/57.jpg)
Gauss Elimination
064.62
30
10
645.700
2.162.60
835
z
y
x
![Page 58: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/58.jpg)
Gauss Elimination
118.8645.7/064.62 z
0502.26
2.6
118.82.1630
y
6413.0
5
118.880502.26310
x
064.62
30
10
645.700
2.162.60
835
z
y
x
![Page 59: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/59.jpg)
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
Forward Elimination
![Page 60: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/60.jpg)
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
0
12 Division By Zero!!Operation Failed
![Page 61: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/61.jpg)
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
12
0OK!!
![Page 62: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/62.jpg)
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
Pivoting
6312 zyx
10830 zyx
![Page 63: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/63.jpg)
Partial Pivoting
nn
nnnnn
lll
n
n
n
b
b
b
b
x
x
x
x
aaaa
aaaa
aaaaaaaa
aaaa
3
2
1
3
2
1
321
ln321
3333231
2232221
1131211
a32>a22
al2>a22
NO
YES
![Page 64: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/64.jpg)
Partial Pivoting
nn
nnnnn
n
n
lll
n
b
b
b
b
x
x
x
x
aaaa
aaaa
aaaaaaaa
aaaa
3
2
1
3
2
1
321
2232221
3333231
ln321
1131211
![Page 65: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/65.jpg)
Full Pivoting
• In addition to row swaping
• Search columns for max elements
• Swap Columns
• Change the order of xi
• Most cases not necessary
![Page 66: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/66.jpg)
EXAMPLE
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
![Page 67: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/67.jpg)
Eliminate Column 1
3
1.0
PIVOTS
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
3.0
1,11
11 ia
apivot i
i
njapivotaa jiijij ,,2,1,11
![Page 68: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/68.jpg)
Eliminate Column 1
6150.70
5617.19
85.7
0200.1019000.00
29333.000333.70
2.01.03
![Page 69: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/69.jpg)
Eliminate Column 2
00333.7
19000.0
PIVOTS
6150.70
5617.19
85.7
0200.1019000.00
29333.000333.70
2.01.03
2,22
22 ia
apivot i
i
njapivotaa jiijij ,,2,1,22
![Page 70: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/70.jpg)
Eliminate Column 2
0843.70
5617.19
85.7
01200.1000
29333.000333.70
2.01.03
UpperTriangular
Matrix[ U ]
ModifiedRHS
{ b }
![Page 71: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/71.jpg)
01200.1000
29333.000333.70
2.01.03
LU DecompositionPIVOTS
Column 1PIVOTS
Column 2
03333.0
1.0 02713.0
![Page 72: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/72.jpg)
LU Decomposition
As many as, and in the location of, zeros
UpperTriangular
MatrixU
01200.1000
29333.000333.70
2.01.03
![Page 73: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/73.jpg)
LU DecompositionPIVOTS
Column 1
PIVOTSColumn 2
LowerTriangular
Matrix
1
1
1
0
0
0
L
03333.0
1.0 02713.0
![Page 74: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/74.jpg)
LU Decomposition
102713.01.0
0103333.0
001
=
This is the original matrix!!!!!!!!!!
01200.1000
29333.000333.70
2.01.03
102.03.0
3.071.0
2.01.03
![Page 75: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/75.jpg)
LU Decomposition
4.71
3.19
85.7
102713.01.0
0103333.0
001
3
2
1
y
y
y
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
[ L ] { y } { b }
[ A ] { x } { b }
![Page 76: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/76.jpg)
LU Decomposition
4.71
3.19
85.7
102713.01.0
0103333.0
001
3
2
1
y
y
y
L y b
85.71 y
5617.190333.03.19 12 yy
0843.70)02713.0(1.04.71 213 yyy
![Page 77: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/77.jpg)
LU Decomposition85.71 y
5617.190333.03.19 12 yy
0843.70)02713.0(1.04.71 213 yyy
0843.70
5617.19
85.7
01200.1000
29333.000333.70
2.01.03
ModifiedRHS
{ b }
![Page 78: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/78.jpg)
LU Decomposition
• Ax=b
• A=LU - LU Decomposition
• Ly=b- Solve for y
• Ux=y - Solve for x
![Page 79: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/79.jpg)
Matrix Inversion
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
bxA
![Page 80: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/80.jpg)
Matrix Inversion
[A] [A]-1
[A] [A]-1=[I]
If [A]-1 does not exist[A] is singular
![Page 81: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/81.jpg)
Matrix Inversion
b xA bxA 1A 1A
I
![Page 82: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/82.jpg)
Matrix Inversion
bAx 1
Solution
![Page 83: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/83.jpg)
Matrix Inversion
[A] [A]-1=[I]
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
![Page 84: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/84.jpg)
Matrix Inversion
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
![Page 85: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/85.jpg)
Matrix Inversion
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
![Page 86: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/86.jpg)
Matrix Inversion
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
![Page 87: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/87.jpg)
Matrix Inversion
• To calculate the invert of a nxn matrix solve n times :
nj
2j
1j
nj
2j
1j
nnn2n1
2n2221
1n1211
a
a
a
aaa
aaa
aaa
nj ,,2,1
otherwise
ji if
0
1ij
![Page 88: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/88.jpg)
Matrix Inversion
• For example in order to calculate the inverse of:
102.03.0
3.071.0
2.01.03
![Page 89: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/89.jpg)
Matrix Inversion
• First Column of Inverse is solution of
0
0
1
a
a
a
102.03.0
3.071.0
2.01.03
31
21
11
![Page 90: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/90.jpg)
Matrix Inversion
0
1
0
a
a
a
102.03.0
3.071.0
2.01.03
32
22
12
• Second Column of Inverse is solution of
![Page 91: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/91.jpg)
Matrix Inversion
• Third Column of Inverse is solution of:
1
0
0
a
a
a
102.03.0
3.071.0
2.01.03
33
23
13
![Page 92: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/92.jpg)
Use LU Decomposition
102713.01.0
0103333.0
001
01200.1000
29333.000333.70
2.01.03
102.03.0
3.071.0
2.01.03
A
![Page 93: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/93.jpg)
Use LU Decomposition – 1st column
• Forward SUBSTITUTION
0
0
1
y
y
y
102713.01.0
0103333.0
001
31
21
11
111 y
03333.00333.00 1121 yy
1009.002713.01.00 211131 yyy
![Page 94: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/94.jpg)
Use LU Decomposition – 1st column
• Back SUBSTITUTION
1009.0
0333.0
1
a
a
a
01200.1000
29333.000333.70
2.01.03
31
21
11
010078.0012.10/1009.0a31
00518.000333.7/a2933.00333.0a 3121
332489.03/a2.0a1.01a 312111
![Page 95: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/95.jpg)
Use LU Decomposition – 2nd Column
• Forward SUBSTITUTION
0
1
0
y
y
y
102713.01.0
0103333.0
001
32
22
12
012 y
122 y
02713.002713.01.00 221232 yyy
![Page 96: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/96.jpg)
Use LU Decomposition – 2nd Column
• Back SUBSTITUTION
02713.0
1
0
a
a
a
01200.1000
29333.000333.70
2.01.03
32
22
12
002709.0012.10/02713.0a32
1429.000333.7/a2933.01a 3222
004944.03/a2.0a1.00a 322212
![Page 97: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/97.jpg)
Use LU Decomposition – 3rd Column
• Forward SUBSTITUTION
1
0
0
y
y
y
102713.01.0
0103333.0
001
33
23
13
013 y
023 y
102713.01.01 231333 yyy
![Page 98: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/98.jpg)
Use LU Decomposition – 3rd Column
• Back SUBSTITUTION
1
0
0
a
a
a
01200.1000
29333.000333.70
2.01.03
33
23
13
09988.0012.10/1a33
004183.000333.7/a2933.00a 3323
006798.03/a2.0a1.00a 332313
![Page 99: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/99.jpg)
Result
102.03.0
3.071.0
2.01.03
A
09988.000271.001008.0
004183.0142903.000518.0
006798.0004944.0332489.0
A 1
![Page 100: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/100.jpg)
Test It
09988.000271.001008.0
004183.0142903.000518.0
006798.0004944.0332489.0
102.03.0
3.071.0
2.01.03
11046.30
1047.31106736.8
0108.11
18
1818
18
![Page 101: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/101.jpg)
Iterative Methods
Recall Techniques for Root finding of Single Equations
Initial Guess
New Estimate
Error Calculation
Repeat until Convergence
![Page 102: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/102.jpg)
Gauss Seidel
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
11
31321211 a
xaxabx
22
32312122 a
xaxabx
33
23213133 a
xaxabx
![Page 103: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/103.jpg)
Gauss Seidel
11
1
11
1312111
00
a
b
a
aabx
22
23112121
2
0
a
axabx
33
1232
113131
3 a
xaxabx
First Iteration: 0,0,0 321 xxx
Better Estimate
Better Estimate
Better Estimate
![Page 104: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/104.jpg)
Gauss Seidel
11
1313
121212
1 a
xaxabx
22
1323
212122
2 a
xaxabx
33
2232
213132
3 a
xaxabx
Second Iteration: 13
12
11 ,, xxx
Better Estimate
Better Estimate
Better Estimate
![Page 105: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/105.jpg)
Gauss SeidelIteration Error:
%1001
, ji
ji
ji
ia x
xx
s
Convergence Criterion:
n
jij
ijii aa1
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
![Page 106: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/106.jpg)
Jacobi Iteration
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
11
31321211 a
xaxabx
22
32312122 a
xaxabx
33
23213133 a
xaxabx
![Page 107: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/107.jpg)
Jacobi Iteration
11
1
11
1312111
00
a
b
a
aabx
22
2321212
00
a
aabx
33
3231313
00
a
aabx
First Iteration: 0,0,0 321 xxx
Better Estimate
Better Estimate
Better Estimate
![Page 108: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/108.jpg)
Jacobi Iteration
11
1313
121212
1 a
xaxabx
22
1323
112122
2 a
xaxabx
33
1232
113132
3 a
xaxabx
Second Iteration: 13
12
11 ,, xxx
Better Estimate
Better Estimate
Better Estimate
![Page 109: ECIV 520 Structural Analysis II Review of Matrix Algebra.](https://reader038.fdocuments.in/reader038/viewer/2022102604/56649d605503460f94a41669/html5/thumbnails/109.jpg)
Jacobi Iteration
Iteration Error:
%1001
, ji
ji
ji
ia x
xx
s