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If we are to have a square matrix (say B), and B = QR, where QR...

If we are to have a square matrix (say B), and B = QR, where QR is the QR factorization of B, Define C = RQ. We need to prove that B and C are similar, and how do we prove C is similar to B assuming that B is symmetric?

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Answer #1

Let us define a symmetric square matrix , where is an orthogonal matrix and is an upper-triangular matrix.

Let . We need to proof is similar to .

Now, since is orthogonal then, .

So,

This is just a similarity transformation of . So we can prove, is similar to

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