I was recently asked to show this for matrix theory class:

Let A be an $ n \times n $ real symmetric matrix such that $ \det(A)\geq 0 $ with n-1 nested principal (not necessarily leading) minors which are positive. We are asked to prove that A is positive semidefinite.

To be honest I know the characterization theorem on positive semidefinite matrices but I cannot see how this leads to an equivalent condition of positive semi definiteness? Maybe the limit of positive semi definite matrices? I therefore ask here, and I thank all helpers.


Relabel the rows and columns so that the nested principal minors are the leading ones. Apply matrix congruence to kill off all but the last entries on the last row/column. The rest is obvious.

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