Question

a) Suppose that A = A^T can be row reduced without row swaps. If E is an elementary matrix such that EA has zero as a second entry in the first column, what can you say about EAE^T?

b) Use step a) to prove that any symmetric matrix that can be row reduced without swaps can be written as A = LDL^T

P.s: L is the lower triangular matrix whose diagonal contains only 1. D is a diagonal matrix whose diagonal contains the pivots of A. Originally, the triangular factorization of A is A = LDU (U is the upper triangular whose diagonal contains only 1), but since A is a symmetric matrix, it can be rewritten as A = LDL^T (U = L^T)

PLEASE HELP ME WITH THIS QUESTION. I HAVE BEEN SPENDING HOURS SOLVING IT AND I GOT STUCK. THANK YOU VERY MUCH FOR YOUR HELP!

Answer 1

Suppose that A = A^T can be row reduced without row swaps. If E is an elementary matrix such that EA has zero as a second entry in the first column, then EA E^T has both the entries (the second entry in the first row and the second entry in the first column ) will be zero. For an illustration take a matrix A which is symmetric, and hence , and E is an elementary matrix which is making the  the second entry in the first column making zero, So after finding EA multiply E^T and we find that the second entry in the first row also becomes zero.

b) Let A be any symmetric matrix. which can be rwo reduced with out swap. This means there exists a set of elementary matrices all of which are of the type (i) adding a scalar multiple of smoe row to another, (ii) multiplying one ro by a non-zero quantity. Thus we have

is an upper triangular matrix. Note that since the lower part of the matrix is made zero, so the product of these elementary matrix is a lower triangular matrix.

Thus we have

As done in part (a) if we multiply the transpose of the elementary matrices from the right side then the corresponding entries of the upper side also ill become zero, thus we obtain

Now note thta we have

Or the inverse of the lower triangular matrix which is again a lower triangular matrix, and thus

Hence the proof.

Hope you have got the idea.

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