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In Zhou's A Practical Guide To Quantitative Finance Interviews I see the following:

A symmetric matrix is positive semidefinite if and only if all its upper left (or lower right) submatrices have nonnegative determinants.

I know that looking only at upper left submatrices (i.e, leading principal minors) is not enough to guarantee semidefinitess (see, e.g., this question).

I'm confused by Zhou's wording. I think it should read "all its upper left and lower right submatrices have nonnegative determinants". Is this a correct statement for semidefiniteness ?
I'm not good enough at linear algebra to come up with a proof.

brised by Linear Algebra
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  • 'I think it should read "all its upper left and lower right submatrices have nonnegative determinants". Is this a correct statement for semidefiniteness ?' No. The correct statement of Sylvester's generalized criterion for semi-definiteness (for real symmetric or hermitian matrixes) is that all principal minors are $\geq 0$. I gave a proof here: https://math.stackexchange.com/questions/4145638/a-is-positive-semidefinite-iff-textdet-b-k-geq-0/ – user8675309 Feb 08 '23 at 17:04

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The statement is false. As a simple counterexample, consider the matrix $$ \pmatrix{0&0&0\\0&-1&0\\0&0&0}. $$ It should be clear that all "leading" and "trailing" principal minors are zero and therefore non-negative. However, this matrix fails to be positive semidefinite.

For an arguably more interesting example, you can consider $$ \pmatrix{ 1&1&1&2\\ 1&1&1&1\\ 1&1&1&1\\ 2&1&1&1 }. $$

Ben Grossmann
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