matrices

matrices - EE103(Fall 2009-10 2 Matrices • definition and...

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Unformatted text preview: EE103 (Fall 2009-10) 2. Matrices • definition and notation • matrix-vector product • matrix-matrix product • matrix inverse • orthogonal matrices • cost of matrix operations 2-1 Matrices m × n-matrix A : A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn • a ij are the elements or coefficients • set of m × n-matrices is denoted R m × n • m , n are the dimensions Special matrices (dimensions follow from context) • A = 0 (zero matrix): a ij = 0 for i = 1 ,...,m , j = 1 ,...,n • A = I (identity matrix): m = n , a ii = 1 , a ij = 0 for i negationslash = j Matrices 2-2 Block matrix notation B = bracketleftbigg 2 2 1 3 bracketrightbigg , C = bracketleftbigg 2 3 5 4 7 bracketrightbigg , D = bracketleftbig 1 bracketrightbig , E = bracketleftbig − 1 6 bracketrightbig A = bracketleftbigg B C D E bracketrightbigg = 2 2 2 3 1 3 5 4 7 1 − 1 6 • A is a block matrix; B , C , D , E are the blocks of A • dimensions of the blocks must be compatible! ( B and D have the same number of columns; B and C have the same number of rows, etc.) Matrices 2-3 Matrix transpose A T = a 11 a 21 ··· a m 1 a 12 a 22 ··· a m 2 . . . . . . . . . . . . a 1 n a 2 n ··· a mn • A T is n × m if A is m × n • ( A T ) T = A • a square matrix A is symmetric if A = A T , i.e. , a ij = a ji Matrices 2-4 Scalar multiplication and addition Scalar multiplication of an m × n-matrix A with a scalar β βA = β a 11 β a 12 ··· β a 1 n β a 21 β a 22 ··· β a 2 n . . . . . . . . . . . . β a m 1 β a m 2 ··· β a mn Addition of two m × n-matrices A and B A + B = a 11 + b 11 a 12 + b 12 ··· a 1 n + b 1 n a 21 + b 21 a 22 + b 22 ··· a 2 n + b 2 n . . . . . . . . . . . . a m 1 + b m 1 a m 2 + b m 2 ··· a mn + b mn Matrices 2-5 Matrix-vector product product of m × n-matrix A with n-vector x Ax = a 11 x 1 + a 12 x 2 + ··· + a 1 n x n a 21 x 1 + a 22 x 2 + ··· + a 2 n x n ....
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This note was uploaded on 10/15/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.

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matrices - EE103(Fall 2009-10 2 Matrices • definition and...

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