# 6_Matrices - A t For a square matrix the main diagonal...

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Matrix multiplication Suppose A is an mxp matrix and B a qxn matrix. The matrix multiplication AB is defined if p=q. The result is a matrix of order mxn. The (i,j)th entry of the result matrix is the dot product of row i of A and column j of B. 9/13/11

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Example
Result matrix size is determined by the number of rows of the first matrix and the number of columns of the second matrix. # of multiplications = 16 Example :

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Matrix multiplication algorithm
Matrix multiplication chain The order of multiplication can cut down computational time. Which is better in terms of computational time? Calculate the complexity of each approach and pick the one with lesser complexity

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Powers of a matrix
Transpose of a matrix Given a matrix A, its transpose it obtained by rewriting rows of A as columns. Transpose of A is denoted by

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Unformatted text preview: A t . For a square matrix, the main diagonal remains the same. The off-diagonal elements are moved to the other side of the diagonal. If A = A t , then A is a symmetric matrix. Determinant of a matrix Deteminants are defined for square matrices. The determinant of a square matrix of order nxn is a function that assigns a scalar value to each possible nxn matrix. If A = (a ij ) nxn , then |A| or det(A) denotes the determinant of A. If |A| = 0, then A is a singular matrix. Laplace Expansion Each element of A, (a ij ), has a minor M ij given by the determinant of the submatrix obtained by removing row i and column j of A. C ij , Cofactor of a ij = (-1) i+j M ij . The matrix of cofactors C=(C ij ) is the adjoint of A. Examples det(A) = det(A t )...
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## This note was uploaded on 03/21/2012 for the course CS 3333 taught by Professor Boppana during the Fall '11 term at The University of Texas at San Antonio- San Antonio.

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6_Matrices - A t For a square matrix the main diagonal...

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