Talk:Sylvester's criterion

Hi, I was wondering if Sylvester's criterion applies to non-symmetric matrices as well. Can I conclude that a non-symmetric matrix is positive definite if the Sylvester's criterion is satisfied?

How does the Sylvester criterion work to show that a matrix is negative defined? I think it is if the principal minors are alternating between negative and positive (<, >, <, >, ...) then the matrix is negative-definite, but it would be nice to have it stated explicitly in the article. — Preceding unsigned comment added by 194.117.40.134 (talk) 15:26, 9 May 2014 (UTC)

_______________________

I point out that there might be a problem with sylvester's criterion as it is stated here regarding the leading principal minors. I found some contribution http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1100319 where it is stated that all principal minors have to be considered, not only the leading ones.

Could someone check this and correct it, if necessary?

--130.83.225.253 (talk) 12:24, 20 February 2015 (UTC)

It is true that for positive semi-definiteness it is needed to check all the principal minors of the matrix, but for positive-definiteness, only the leading principal minors need to be checked. Saung Tadashi (talk) 13:24, 20 February 2015 (UTC)

The "the proof of the general case" section

This section is hard to understand, and doesn't seem to end up proving anything except 'the "only if" part of Sylvester's Criterion for non-singular real-symmetric matrices.' But this was the easy part to begin with.

Nowhere does it address the positive semidefinite case, where A is singular.

I propose the section be deleted, the first section be renamed "proof in the case of strictly positive leading principal minors." 99.239.96.82 (talk) 14:48, 15 September 2023 (UTC)

I deleted this section and retitled the first, which contains a complete proof for the PD case.205.175.106.80 (talk) 20:26, 5 February 2024 (UTC)