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LU Decomposition (i). Prove that for n equal to 2 or 3 there is a non-singular square...

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Advance Math

LU Decomposition

(i). Prove that for n equal to 2 or 3 there is a non-singularsquare (n by n) matrix which has no LU decomposition with L unitlower triangular and U upper triangular. (In fact, this is true forany integer ? 2.)

(ii). We will see that all non-singular square matrices do havean LUP decomposition (some time soon in class). Here P is apermutation matrix, also defined in Appendix D and used in Chapter28.

Show that the inverse of a permutation matrix P is also apermutation matrix. (You can do this by explicitly defining what P?1 when P is given.)

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