Modified Gary Larson Far Side cartoon Matlab Linear

Modified Gary Larson Far Side cartoon

Matlab Linear Algebra Review

Matrices can represent sets of equations! a 11 x 1+a 12 x 2+…+a 1 nxn=b 1 a 21 x 1+a 22 x 2+…+a 2 nxn=b 2 … am 1 x 1+am 2 x 2+…+amnxn=bm What’s the matrix representation?

Vectors

Matrices U = [umn] = u 11 u 21 um 1 u 12 u 22 … um 2 … … u 1 n u 2 n … 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 X

• 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 w The inner product of vector multiplication is a SCALAR

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 / indicates that 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 = The outer product of vector multiplication is a MATRIX 24 6 30 8 2 10 -12 -3 -15

Matrix Multiplication The inner numbers have to match 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.

B a 21 C A a 22 In MATLAB, the * symbol represents matrix multiplication : >>A=B*C

• Matrix multiplication is not commutative! C*B #taken from previous slide ans = 19 18 17 16 24 32 32 24 16 • Matrix multiplication is distributive and associative A(B+C) = AB + BC (AB)C = A(BC)

Revisit the vector example >>a'*b >>a*b’ ans = 11 24 6 30 8 2 10 -12 -3 -15 a 1 x 3 * b 3 x 1 = c 1 x 1 a 3 x 1 * b 1 x 3 = c 3 x 3

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 function For 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) = a 11 a 22 a 33+ a 12 a 23 a 31 + a 13 a 21 a 32 − a 11 a 23 a 32 − a 12 a 21 a 33 − a 13 a 22 a 31

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 direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. c = a 2 b 3 -a 3 b 2 = 2*5 – (-3)*1 = 13 a 3 b 2 -a 1 b 3 -3*4 – 6*5 -42 a 1 b 2 -a 2 b 1 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 24 24 -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 - 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 Regular matrix multiplication – in this case with vectors 1 x 2 * 2 x 1 = 1 x 1 => 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 2 x 1 * 1 x 2 = 2 x 2 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 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, AB 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 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 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 a b c 2 a + 1 b = c

Linear Independence a b c There is no simple, linear equation 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