Modified Gary Larson Far Side cartoon
description
Transcript of Modified Gary Larson Far Side cartoon
Modified Gary Larson Far Side cartoon
Matlab
Linear Algebra Review
Matrices can represent sets of equations!
a11x1+a12x2+…+a1nxn=b1
a21x1+a22x2+…+a2nxn=b2
…
am1x1+am2x2+…+amnxn=bm
What’s the matrix representation?
€
A =
a11 a12 ... a1na21 a22 ... a2n
...am1 am2 ... amn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
x =
x1x2.xn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
b =
b1b2.bm
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
Ax = b
Vectors
Matrices
U = [umn] =
u11 u12 … u1n
u21 u22 … u2n
…um1 um2 …
umn
The general matrix consists of m rows and n columns. It is also known as an m x n (read m by n) array.
Each individual number, uij, of the array is called the element
Elements uij where i=j is called the principal diagonal
Transpose of a Matrix
Matrix & Vector Addition
Vector/Matrix addition is associative and commutative(A + B) + C = A + (B + C); A + B = B + A
Matrix and Vector Subtraction
Vector/Matrix subtraction is also associative and commutative(A - B) - C = A - (B - C); A - B = B - A
Matrix and Vector Scaling
X
ax
• For addition and subtraction, the size of the matrices must be the same
Amn + Bmn = Cmn
• For scalar multiplication, the size of Amn does not matter
• All three of these operations do not differ from their ordinary number counterparts
• The operators work element-by-element through the array, aij+bij=cij
Vector Multiplication
• The inner product or dot product
v
wa
€
v ⋅w = (x1,x2) ⋅(y1,y2) = x1y1 + x2y2
The inner product of vector multiplicationis a SCALAR
€
v ⋅w = (x1,x2) ⋅(y1,y2) =||v ||⋅ ||w || cosα
Inner product represents a row matrix multiplied by a column matrix. A row matrix can be multiplied by a column matrix, in that order, only if they each have
the same number of elements!In MATLAB, in order to properly calculate
the dot product of two vectors use
>>sum(a.*b)
element by element multiplication (.*)sum the results
A . prior to the * or / indicatesthat matlab should perform the array or element by element calculation rather
than linear algebra equivalent
or
>>a’*b
• The outer product
A column vector multiplied by a row vector.
In Matlab:
>>a*b’
ans =
24 6 30 8 2 10 -12 -3 -15
The outer product of vector multiplicationis a MATRIX
Two matrices can be multiplied together if and only if
the number of columns in the first equals the number of rows in the second.
Matrix Multiplication
pmmnpn BAC
m
kkjikij bac
1
The inner numbers have to match
a21 a22
In MATLAB, the * symbol represents matrix multiplication :
>>A=B*C
B C A
C*B #taken from previous slide ans =
19 18 17
16 24 32
32 24 16
nnnnnnnn ABBA
• Matrix multiplication is not commutative!
• Matrix multiplication is distributive and associative
A(B+C) = AB + BC
(AB)C = A(BC)
Revisit the vector example
>>a'*b
ans =
11
a1x3 * b3x1 = c1x1
>>a*b’
ans =
24 6 30 8 2 10 -12 -3 -15
a3x1 * b1x3 = c3x3
Dot products in Matlab(using this form – built in functions - don’t have to match dimensions of vectors – can mix
column and row vectors – although they have to be the same length)
>> a=[1 2 3];>> b=[4 5 6];
>> c=dot(a,b)c = 32
>> d=dot(a,b’)d = 32
Dot products using built-in functionFor matrices – does dot product of columns.
Essentially treating the columns like vectors.The matrices have to be the same size.
>> a=[1 2;3 4]a = 1 2 3 4
>> b=[5 6;7 8]b = 5 6 7 8
>> dot(a,b)ans = 26 44
Determinant of a Matrix
or follow the diagonals
det (A) = a11a22a33+ a12a23a31 + a13a21a32 − a11a23a32 − a12a21a33 − a13a22a31
Cross-product of two vectorsThe cross product a × b is defined as a vector c that is
perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
€
a × b = a b sinθ
c = a2b3-a3b2 = 2*5 – (-3)*1 = 13
a3b2-a1b3 -3*4 – 6*5 -42
a1b2-a2b1 6*1 – 2*4 -2
Cross products in Matlab(using this form – built in functions - don’t have to match dimensions of vectors – can mix
column and row vectors – although they have to be the same length)
>> a=[1 2 3];>> b=[4 5 6];
>> e=cross(a,b)e = -3 6 -3
>> f=cross(a,b’)f = -3 6 -3
>> g=cross(b,a)g = 3 -6 3
For matrix – does cross product of columns. (one of the dimensions has to be 3 and takes other dimension as additional vectors)
>> a=[1 2;3 4;5 6]a = 1 2 3 4 5 6
>> b=[7 8;9 10;11 12]b = 7 8 9 10 11 12
>> cross(a,b)ans = -12 -12 24 24 -12 -12
Matrix Operators
• + Addition• - Subtraction• * Multiplication• / Division• \ Left division • ^ Power• ' Complex conjugate transpose• ( ) Specify evaluation order
Array Operators
• + Addition• - Subtraction• .* Element-by-element multiplication• ./ Element-by-element division.
• A./B: divides A by B by element
• .\ Element-by-element left division• A.\B divides B by A by element
• .^ Element-by-element power• .' Unconjugated array transpose
• the signs of imaginary numbers are not changed, unlike a regular matrix transpose
Operators as built-in commands
plus - Plus + uplus - Unary plus + minus - Minus - uminus - Unary minus - mtimes - Matrix multiply * times - Array multiply .* mpower - Matrix power ^ power - Array power .^ mldivide - Backslash or left matrix divide \ mrdivide - Slash or right matrix divide / ldivide - Left array divide .\ rdivide - Right array divide ./ cross - cross product
Multiplication in Matlab
>> x=[1 2];>> y=[3 4];
>> z=x*y’z = 11
>> w=x.*yw = 3 8
>> z=x'*yz = 3 4 6 8
Regular matrix multiplication – in this case with vectors 1x2 * 2x1 = 1x1 => dot productElement by element multiplication
Regular matrix multiplication – in this case with vectors 2x1 * 1x2 = 2x2 matrix
Division in MatlabIn ordinary math, division (a/b) can be thought of as a*1/b or a*b-1.
A unique inverse matrix of B, B-1, only potentially exists if B is square. And matrix multiplication is not communicative, unlike ordinary multiplication.
There really is no such thing as matrix division in any simple sense.
/ : B/A is roughly the same as B*inv(A). A and B must have the same number of columns for right division.
\ : If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way. A and B must have the same number of rows for left division.
If A is an m-by-n matrix (not square) and B is a matrix of m rows, AX=B is solved by least squares.
The / and \ are related
B/A = (A'\B')’
Inverse of a Matrix
the determinant
the principal diagonal elements switch
the off diagonal elements change sign
• Square matrices with inverses are said to be nonsingular
• Not all square matrices have an inverse. These are said to be singular.
• Square matrices with determinants = 0 are also singular.
• Rectangular matrices are always singular.
Right- and Left- Inverse
If a matrix G exists such that GA = I, than G is a left-inverse of A
If a matrix H exists such that AH = I, than H is a right-inverse of A
Rectangular matrices may have right- or left- inverses, but they are still singular.
Some Special Matrices
• Square matrix: m (# rows) = n (# columns)
• Symmetric matrix: subset of square matrices where AT = A
• Diagonal matrix: subset of square matrices where elements off the principal diagonal are zero, aij = 0 if i ≠ j
• Identity or unit matrix: special diagonal matrix where all principal diagonal elements are 1
Linear Dependence
2a + 1b = c
a b c
Linear Independence
There is no simple, linear equation that can make these vectors related.
a b c
Rank of a matrix
Acknowledgement
• This lecture borrows heavily from online lectures/ppt files posted by
• David Jacobs at Univ. of Maryland• Tim Marks at UCSD• Joseph Bradley at Carnegie Mellon