Is this matrix positive definite (are eigenvalues always positive)?

Question

I am not a mathematician, could you please tell me whether,

$(\Sigma_A) * (\Sigma_B)^{-1} $ is positive-definite or not?

Where $\Sigma_A$ and $\Sigma_B$ are covariance matrices of A and B. And, "A" and "B" are two real general matrices with dimensions [mxn].

I am programming something and it is important to know whether the eigenvalues of the above multiplications are always positive or not.

score 2 · Accepted Answer · answered Dec 26 '16 at 18:45

2

$\Sigma_A \Sigma_B^{-1}$ will not always be positive definite, since it will not generally be symmetric. However, it will have positive eigenvalues, and these eigenvalues are the same as those of the matrix $\Sigma_B^{-1/2}\Sigma_A\Sigma_B^{-1/2}$, which is positive definite.

answered Dec 26 '16 at 18:45

Ben Grossmann

225,327

Sometimes one talks about unsymmetric positive definite systems, tho (e.g. this paper. – J. M. ain't a mathematician Dec 26 '16 at 20:04
@J.M. sure, but covariance matrix are specifically of the form $\Sigma_A = AA^T$ (or sometimes $A^TA$). – Ben Grossmann Dec 26 '16 at 20:11
1

I'm well aware, so the product is certainly not SPD. Just saying that "positive definite" doesn't always include "symmetric" as a condition. – J. M. ain't a mathematician Dec 26 '16 at 20:17
@J.M. Ah, I see now. If I remember correctly, there exists an example of symmetric positive definite matrices $A,B$ such that the product $AB$ fails to be (unsymmetric) positive definite. – Ben Grossmann Dec 26 '16 at 20:29
@J.M. see this answer in particular – Ben Grossmann Dec 26 '16 at 20:32
Just found that answer too. :) – J. M. ain't a mathematician Dec 26 '16 at 20:33

Is this matrix positive definite (are eigenvalues always positive)?

1 Answers1