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Modified Gary Larson Far Side cartoon Matlab Linear Algebra Review Matrices can represent sets of equa>ons! a11x1+a12x2+…+a1nxn=b1 a21x1+a22x2+…+a2nxn=b2 … am1x1+am2x2+…+amnxn=bm What’s the matrix representa>on? ⎡ x1⎤
⎡ a11 a12 ... a1n ⎤
⎢ ⎥
⎢
⎥
x2⎥
a21
a22
...
a2n
⎢
⎢
⎥
x=
A=
⎢ . ⎥
⎢
⎥
...
⎢ ⎥
⎢
⎥
⎣ xn⎦
⎣am1 am2 ... amn ⎦
⎡ b1 ⎤
⎢ ⎥
b2 ⎥
⎢
b=
⎢ . ⎥
⎢ ⎥
⎣bm ⎦
Ax = b
Vectors Matrices u11 u21 U = [umn] = um1
u12 … u1n u22 … u2n … 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 Addi>on Vector/Matrix addi>on is associa>ve and commuta>ve (A + B) + C = A + (B + C); A + B = B + A Matrix and Vector Subtrac>on Vector/Matrix subtrac>on is also associa>ve and commuta>ve (A -­‐ B) -­‐ C = A -­‐ (B -­‐ C); A -­‐ B = B -­‐ A Matrix and Vector Scaling •  For addi>on and subtrac>on, the size of the matrices must be the same Amn + Bmn = Cmn •  For scalar mul>plica>on, the size of Amn does not ma^er •  All three of these opera>ons do not differ from their ordinary number counterparts •  The operators work element-­‐by-­‐element through the array, aij+bij=cij Vector Mul>plica>on •  The inner product or dot product v ⋅ w = (x1, x 2 ) ⋅ (y1, y 2 ) = x1 y1 + x 2 y 2
€
The inner product of vector mul>plica>on is a SCALAR v ⋅ w = (x1, x 2 ) ⋅ (y1, y 2 ) =|| v ||⋅ || w || cos α
Inner product represents a row matrix mul>plied by a column matrix. A row matrix can be mul>plied 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 mul>plica>on (.*) sum the results A . prior to the * or / indicates that matlab should perform the array or element by element calcula>on rather than linear algebra equivalent or >>a’*b •  The outer product A column vector mul>plied by a row vector. In Matlab: >>a*b’ ans = The outer product of vector mul>plica>on is a MATRIX 24 6 30 8 2 10 -­‐12 -­‐3 -­‐15 Matrix Mul>plica>on The inner numbers have to match Two matrices can be mul>plied together if and only if the number of columns in the first equals the number of rows in the second. B a21 C A a22 In MATLAB, the * symbol represents matrix mul>plica>on : >>A=B*C •  Matrix mul>plica>on is not commuta>ve! C*B #taken from previous slide ans = 19 18 17 16 24 32 32 24 16 •  Matrix mul>plica>on is distribu>ve and associa>ve A(B+C) = AB + BC (AB)C = A(BC) Revisit the vector example >>a'*b >>a*b’ ans = ans = 11 24 6 30 8 2 10 -­‐12 -­‐3 -­‐15 a1x3 * b3x1 = c1x1 a3x1 * b1x3 = c3x3 Dot products in Matlab (using this form – built in func>ons -­‐ 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 func>on For matrices – does dot product of columns. Essen>ally trea>ng 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 vectors The cross product a × b is defined as a vector c that is perpendicular to both a and b, with a direc>on given by the right-­‐hand rule and a magnitude equal to the area of the parallelogram that the vectors span. a × b = a b sin θ
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 func>ons -­‐ 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 addi>onal 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 • 
• 
• 
• 
• 
• 
• 
• 
+ Addi>on -­‐ Subtrac>on * Mul>plica>on / Division \ Let division ^ Power ' Complex conjugate transpose ( ) Specify evalua>on order Array Operators • 
• 
• 
• 
+ Addi>on -­‐ Subtrac>on .* Element-­‐by-­‐element mul>plica>on ./ Element-­‐by-­‐element division. •  A./B: divides A by B by element •  .\ Element-­‐by-­‐element let 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
+
+
*
.*
^
.^
\
/
.\
./
Mul>plica>on in Matlab >> x=[1 2]; >> y=[3 4]; >> z=x*y’ z = 11 Regular matrix multiplication – in this case
with vectors 1x2 * 2x1 = 1x1 => dot product
>> w=x.*y w = 3 8 Element by element multiplication
>> z=x'*y z = 3 4 6 8 Regular matrix multiplication – in this case
with vectors 2x1 * 1x2 = 2x2 matrix
Division in Matlab In 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 poten>ally exists if B is square. And matrix mul>plica>on is not communica>ve, unlike ordinary mul>plica>on. 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 let 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 principal diagonal elements switch the off diagonal elements change sign the determinant •  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 Let-­‐ Inverse If a matrix G exists such that GA = I, than G is a let-­‐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 let-­‐ inverses, but they are s>ll 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 •  Iden>ty or unit matrix: special diagonal matrix where all principal diagonal elements are 1 Linear Dependence a b c 2a + 1b = c Linear Independence a b c There is no simple, linear equa>on that can make these vectors related. 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