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 A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 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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matrices - EE103 (Fall 2009-10) 2. Matrices definition and...

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