Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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If A is a (rectangular) n × m matrix, it can be shown that the eigenvalues of the symmetric square matrix AT A are non-negative. The singular values 01,02,...,0m of A are then defined to be the positive square roots of the eigenvalues of AT A. It can also be shown: The rank of a matrix is equal to the number of its non-zero singular values. Confirm this for the following matrices: A = D - 0 0 1 0 0100 1000 1 1 1 B= 1 0 0 0 0100 0010 0001 0 000 C = (13), 1 2 2 F-( 14 ) -(13) E = F = −1 0 0 Hint: The elementary unit vectors e; are the eigenvectors of a diagonal ma- trix.
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Transcribed Image Text:If A is a (rectangular) n × m matrix, it can be shown that the eigenvalues of the symmetric square matrix AT A are non-negative. The singular values 01,02,...,0m of A are then defined to be the positive square roots of the eigenvalues of AT A. It can also be shown: The rank of a matrix is equal to the number of its non-zero singular values. Confirm this for the following matrices: A = D - 0 0 1 0 0100 1000 1 1 1 B= 1 0 0 0 0100 0010 0001 0 000 C = (13), 1 2 2 F-( 14 ) -(13) E = F = −1 0 0 Hint: The elementary unit vectors e; are the eigenvectors of a diagonal ma- trix.
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