Numerical sensitivity of X'*Y*X given X is a permutation matrix

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Assume X is an arbitrary matrix and Y is a symmetric matrix with double floating-point format. I understand that the product X'*Y*X can become asymmetric due to the floating point numerics problem.
However, I am not sure this potential 'asymmetric' problem would happen if X here is assumed to be a permutation matrix. I would think the structure of a permutation matrix may not cause any problem to the symmetric structure of X'*Y*X
If anyone has suggestions, please share them.
  2 Comments
David Goodmanson
David Goodmanson on 31 Mar 2017
Hi Qian, If your matrices are real then M' is the same as the transpose M.', so the statement about being symmetric makes sense. If anything were complex then you would have to make different statements about X' *Y*X and X.' *Y*X . That nitpick aside, for matrix multiplication by permutation matrices all you are doing is multiplying by 1 and adding 0, and it's hard to see how that could have any bad effect numerically.

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Accepted Answer

John D'Errico
John D'Errico on 31 Mar 2017
Edited: John D'Errico on 31 Mar 2017
Not a true answer, but IF it were going to happen, you would see it here:
[Q,R,X] = qr(rand(3000));
Y = randn(3000);
Y = Y + Y';
Z = X'*Y*X;
all(all((Z-Z') == 0))
ans =
logical
1
Often a simple test is worth a lot, just to verify intuition.
So X is a random permutation matrix. Y a random symmetric matrix. Z is perfectly symmetrical. I chose a large enough matrix so that it had 4 CPUs running on the problem, so the BLAS will be kicking in. Usually when there are problems on something like this, it is the BLAS that are claimed to be the culprit.
I think the important point is (I see David made this in his comment) that there are NO adds involved between two non-zero numbers. All of the adds will always end up being of the form u+0 or 0+u, where there can never be any kind of error introduced into the least significant bits.
So I'll claim that this operation, where Y is symmetric and X a permutation matrix will never introduce an asymmetry into the result.

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