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Unformatted text preview: Review of Elementary Matrix Algebra: Lecture XXVI Charles B. Moss November 9, 2010 Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 1 / 31 1 Examples Inverse Rank of a Matrix 2 Vector Space Linear Independence and Dependenc Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 2 / 31 Basic Definitions 1. A matrix A of size m n is an m n rectangular array of scalars A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn (1) It is sometimes useful to partition matrices into vectors. A = a 1 a 2 a n a 1 = a 11 a 21 . . . a m 1 a n = a 1 n a 2 n . . . a mn (2) Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 3 / 31 Continued A = a 1 a 2 . . . a m a 1 = a 11 a 12 a 1 n a 2 = a 21 a 22 a 2 n . . . a m = a m 1 a m 2 a mn (3) The sum of two identically dimensioned matrices can be expressed as A + B = [ a ij + b ij ] (4) Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 4 / 31 In order to multiply a matrix by a scalar, multiply each element of the matrix by the scalar. In order to discuss matrix multiplication, we first discuss vector multiplication. Two vectors x and y can be multiplied together to form z ( z = x y ) only if they are conformable. If x is of order 1 n and y is of order n 1, then the vectors are conformable and the multiplication becomes z = xy = n X i =1 x i y i (5) Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 5 / 31 Extending this discussion to matrices, two matrices A and B can be multiplied if they are conformable. If A is order k n and B is of order n 1 then the matrices are conformable. Using the partitioned matrix above, we have C = AB = a 1 a 2 . . . a k b 1 b 2 b l = a 1 b 1 a 1 b 2 a 1 b l a 2 b 1 a 2 b 2 a 2 b l . . . . . . . . . . . . a k b 1 a k b 2 a k b l (6) Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 6 / 31 Theorem 1.1 Let and be scalars and A , B , and C be matrices. Then when the operations involved are defined, the following properties hold I A + B = B + A . I ( A + B ) + C = A + ( B + C ) I ( A + B ) = A + B I ( + ) = A + B I A A = A + ( A ) = [0] I A ( B + C ) = AC + BC I ( A + B ) C = AC + BC I ( AB ) C = A ( BC ) Charles B. Moss () Multivariate Hypothesis Testing November 9, 2010 7 / 31 The transpose of an m n matrix is a n m matrix with the rows and columns interchanged. The transpose of A is denoted A ....
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
 Spring '10
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