Numerical sensitivity of function kron(X,Y)
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Assume X and Y are both symmetric matrices with double floating-point format. Can kron(X,Y) potentially become asymmetric due to the floating point numerics similar to the situation like X'*Y*X ?
I personally think it would be quite unlikely to have the problem I stated above given the mathematical definition of the Kronecker product. Nevertheless, if anyone has suggestions, please share them.
Matt J on 19 Dec 2016
Edited: Matt J on 19 Dec 2016
My instinct and my tests say no. There are only scalar multiplications in A=kron(X,Y), so those should be pretty safe. However, if you plan to start multiplying A with other data, the benefits of the symmetry might go out the window pretty fast. Note, for example, that C as computed with,
C=reshape( kron(X,X)*Y(:) ,N,N);
should be symmetric in exact arithmetic for symmetric X and Y. However, I find that this fails in most numerical tests, likely due to the very same issue you pointed out with X.'*Y*X.
More Answers (1)
John D'Errico on 19 Dec 2016
Never trust the LSBs in a floating point computation. If it can go bad, it probably will.
A = randn(200);
B = randn(100);
A = A + A';
B = B + B';
C = kron(A,B);
all(all(C == C'))
all(all(C == C'))
I did my best above, but symmetry remains. C is pretty large, enough so that the blas should be kicking in, just in case they might have gotten me in trouble. I even chose different sizes for A and B, in case that might get me in trouble.
So I did a quick look at the guts of kron. It looks like it is safe to retain symmetry to me. A computation like X'*Y*X is a problem, because first MATLAB forms X'*Y, which is not symmetric, even if X and Y are symmetric. Then, even though a final multiplication by X should restore symmetry, that simply won't happen in floating point arithmetic.
Though still a conjecture and not proved, I think kron is ok.