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)[reply]

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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)[reply]

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)[reply]

The "the proof of the general case" section[edit]

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)[reply]

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)[reply]