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Assume a real square non-symmetric matrix $M$ is given and the task is to analyse if the matrix is positive definite, that is if $x^TMx>0$.

Is it possible to make assertions about definiteness by only checking the principle minors?

That means, the analysis should not necessarily include a consideration of the symmetric form $(M+M^T)/2$.

Carlos
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  • you need at least that all eigenvalues are positive – Surb May 01 '22 at 22:03
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    @Surb Definiteness is not characterised by eigenvalues in the non symmetric case: https://math.stackexchange.com/q/83134/207930 – Carlos May 01 '22 at 22:54

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It is not clear how to adapt the usual Sylverster's criterion to non-symmetric matrices. For instance, consider the matrix $$ M= \begin{pmatrix} 1 & a \\ b & 1 \end{pmatrix}, $$ where $a$ and $b$ will be fixed later. The principal minors are $1$ and $1-ab$. But for the symmetric matrix $$ \frac{M+M^T}{2} = \begin{pmatrix} 1 & (a+b)/2 \\ (a+b)/2 & 1 \end{pmatrix}, $$ the principal minors are $1$ and $1-((a+b)/2)^2$.

This way, the map $x \mapsto x^t M x$ is positive definite if and only if $1-((a+b)/2)^2>0$. This is not something you can check only looking at $1-ab$. For instance, if $a=0, b=10$, then both $1$ and $1-ab$ are positive, but $x \mapsto x^T M x$ is not positive definite.

Nicolás Vilches
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  • This is a really interesting example. I only found this paper of Fiedler and Ptak https://link.springer.com/article/10.1007/BF02166034 , though, I cannot find out so far the relation of positive definiteness and the principal minors. – Carlos May 01 '22 at 22:47