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Introduction to Matrices and Vectors Sebastian van Delden USC Upstate [email protected] Introduction Definition: A matrix is a rectangular array of numbers. a11 a12 a a22 21 A a m1 am 2 a1n a2 n aij amn element in ith row, jth column m rows Also written as A=aij mn matrix n columns When m = n, A is called a square matrix. Matrix Equality Definition: Let A and B be two matrices. These matrices are the same, that is, A = B if they have the same number of rows and columns, and every element at each position in A equals the element at corresponding position in B. * This is not trivial if elements are real numbers subject to digital approximation. The Transpose of a Matrix x11 x12 x x22 21 T X xn1 xn 2 x11 x21 x x22 12 x1n x2 n x1m x2 m xnm xm1 xm 2 xmn T 1 2 3 A 4 5 6 1 4 A T 2 5 3 6 Note that (XT)T = X 14 2 B 3 14 14 3 B 2 14 T Matrix Addition, Subtraction Let A = aij , B = bij be mn matrices. Then: A + B = aij + bij, and A – B = aij – bij 1 1 3 4 4 3 3 4 1 4 4 0 2 0 2 3 4 3 1 1 3 4 2 5 3 4 1 4 2 8 2 0 2 3 0 3 5 Properties of Matrix Addition Commutative: A+B=B+A Associative: A + (B + C) = (A + B) + C Inventories Makealot, Inc. manufactures widgets, nerfs, smores, and flots. It supplies three different warehouses (#1,#2,#3). Opening inventory: w Sales: n s f #1 #2 20 50 25 10 #3 2 55 12 33 90 45 89 Closing inventory: 6 – 15 0 25 3 5 50 0 7 35 6 30 10 = 20 6 3 80 2 4 12 41 77 3 3 0 Scalar Multiplication 1 3 10 5 7 2 4 10 30 6 50 8 70 20 40 60 80 Associative: c1(c2A) = (c1c2)A Distributive: (c1 + c2) A = c1A + c2A Matrix Multiplication Let A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij] k cij ait btj ai1b1 j ai 2b2 j t 1 a11 a 21 a12 a22 a13 a23 b11 b12 b b 21 22 b31 b32 b13 b23 b33 aik bkj b14 c11 c12 b24 c21 c22 b34 a11b12 a12b22 a13b32 c12 c13 c23 c14 c24 Matrix Multiplication Let A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij] k cij ait btj ai1b1 j ai 2b2 j t 1 a11 a 21 a12 a22 a13 a23 b11 b12 b b 21 22 b31 b32 b13 b23 b33 aik bkj b14 c11 c12 b24 c21 c22 b34 a21b12 a22b22 a23b32 c22 c13 c23 c14 c24 Matching Dimensions To multiply two matrices, the dimensions must match: 23 34 24 matrix have to be equal a11 a 21 a12 a22 23 a13 a23 b11 b12 b b 21 22 b31 b32 b13 b23 b33 34 b14 c11 c12 b24 c21 c22 b34 c13 c23 24 8 dot products c14 c24 Multiplicative Properties Note even if AB is defined, BA might not be. Example: If A is 34, B is 46, then AB is a 36 matrix, but BA is not defined. Even if both AB and BA are defined, they may not have the same dimensions. Even if they do, the result might not be equal. However, provided that the dimensions match, (AB)C = A(BC) 1 1 2 1 A B 2 1 1 1 3 2 4 3 AB BA 5 3 3 2 Chained Matrix Multiplication What is the most efficient way of carrying out the following chained matrix multiplication? M 135 589 893 334 Chained Matrix Multiplication What is the most efficient way of carrying out the following chained matrix multiplication. M M 1 135 M 2 589 M 3 893 M 4 334 Let's try: M M 1 135 5 89 3 133553 M 4 334 M 13 5 3 195133 M 4 334 M 13 3 34 13261334 Total multiplications = 1335 195 1326 2856. Chained Matrix Multiplication What is the most efficient way of carrying out the following chained matrix multiplication. M M 1 135 M 2 589 M 3 893 M 4 334 Answer: M M 1 135 M 2 589 M 3 893 M 4 334 This would require 2856 multiplications Example What is the most efficient way of carrying out the following chained matrix multiplication? M 95 52 26 Example What is the most efficient way of carrying out the following chained matrix multiplication? M 95 52 26 Answer: If we do 95 52 26 the cost is 5 2 6 9 5 6 330 If we do 95 52 26 the cost is 9 5 2 9 2 6 198. So, this is the optimal way. Ways of Parenthesizing a product of n matrices Let T(n) be the number of essentially distinct ways of parenthesizing a product of n matrices. The values of T(n) are known as Catalan numbers. Here are few values of T(n): n 1 2 3 4 5 … 10 … 15 T(n) 1 1 2 5 14 … 4862 … 2674440 It can be shown that T(n) = Ω(22n/n2) Identity Matrix The identity matrix is a square matrix with all 1’s along the diagonal and 0’s elsewhere. Example: 1 0 0 For an mn matrix A, Im A = A In I3 0 1 0 (mm) (mn) = (mn) (nn) 0 0 1 1 0 0 a 0 1 0 c 0 0 1 e b a 0 0 b 0 0 d 0 c 0 0 d 0 f 0 0 e 0 0 f Inverse Matrix Let A and B be nn matrices. If AB=BA=In then B is called the inverse of A, denoted B=A-1. Not all square matrices are invertible. Symmetric Matrix If matrix A is such that A = AT then it is called a symmetric matrix. For example: 1 4 1 4 3 0 1 0 2 is symmetric. Note, for A to be symmetric, is has to be square. Note also that In is trivially symmetric. Vectors An m element column vector a 1 a a 2 a m Transpose the column aT [a1 a2 am ]. A q element row vector b [b1b2 bq ] Transpose the row b1 b 2 T b bq Vectors A 1xN or Nx1 matrix 1xN is called a row vector Nx1 is called a column vector N is the dimension of the vector Vectors can be drawn as arrows and so have a direction and a magnitude. Magnitude: a12 a22 ... an2 a1 a given a 2 an Drawing Vectors y a = (8,5) 5 8 x Unit Vectors Magnitude is 1 A normalized vector is a unit vector that has be obtained by divided each dimension of a vector by its magnitude. It has the same direction as the original vector. Important because something direction is all that is important – magnitude is not needed… y |a| = sqrt(82 + 52) =~ 9.4 Normalized a, a’ = (8/9.4, 5/9.4) = (.85, .53) 5 a = (8,5) a’ = (.85, .53) 8 x Geometry of Vectors If m is magnitude: a = m . cos y , b = m . sin For unit vectors: a = cos , b = sin (a, b) b = tan-1(b/a) a x Addition - preserves direction and magnitude. - application: robot position translations - tip to tail method: y u+v u v x Subtraction - application: can represent robot position error vector - u – v, a vector originating in v and ending in u y u-v u v x Multiplication with a scalar - can change magnitude and direction (if multiplied with a negative number. y u v ½v x -u Cross Product Produces a vector perpendicular (normal) to the plane created by the 2 vectors. uxv uxv v u Cross product Direction is determined by the right hand rule Put hand on first vector (left side of x) and curl fingers towards second vector. Magnitude of u x v is |u| . |v| . sin(theta) where theta is the angle between u and v So, cross product produces a vector Dot product Length of the projection of one vector onto a another. u . v Dot product is |u| . |v| . cos(theta) where theta is the angle between u and v So, dot product produces a scalar Note: is u and v and unit vectors, the dot product is simply: cos(theta) u theta cos (theta) v Dot and cross products Dot product from unit vectors: As angle approaches 0, dot product approaches 1 As angle approaches 90, dot product approaches 0 Cross product from unit vectors: As angle approaches 0, dot product approaches 0 As angle approaches 90, dot product approaches 1 Finally…. Perpendicular vectors (dot product = 0) are called orthogonal vectors. Orthogonal unit vectors are called orthonormal vectors. Think: what do you need to represent a 3D coordinate system…? Three orthonormal vectors: X, Y, and Z….