SolSec 4.9 - Problems and Solutions for Section 4.9(4.72...

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Problems and Solutions for Section 4.9 (4.72 through 4.82) 4.72 Consider the mass matrix M = 10 1 1 1 and calculate M -1 , M -1/2 , and the Cholesky factor of M . Show that LL M M M I M M M T = = = 1 2 1 2 1 2 1 2 / / / / Solution: Given M = 10 1 1 1 The matrix, P , of eigenvectors is P = 0 1091 0 9940 0 9940 0 1091 . . . . The eigenvalues of M are λ λ 1 2 0 8902 10 1098 = = . . From Equation M P P M T = = 1 1 2 1 1 1 0 1111 0 1111 0 1111 1 1111 diag λ λ , , . . . . From Equation M Vdiag V M T = [ ] = 1 2 1 1 2 2 1 2 1 2 0 3234 0 0808 0 0808 1 0510 / / / / , . . . . λ λ The following Mathcad session computes the Cholesky decomposition.
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