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I wanted to ask why is positive definite semi-ordering is well defined only for Hermitian matrices (or symmetric matrices if restricted to the reals)?

I saw an extension to the definition of positive definite matrices to the asymmetrical case. Namely, we say that a real (unnecessary symmetric) matrix $M$ is positive definite in the wide sense if and only if $z^TMz>0$ for all non-zero vectors $z$. This is equivalent to that the symmetrical part of $M$, which is $\frac{M+M^T}{2}$, is positive definite in the narrow sense.

Why is not possible to define a semi-ordering using this extended definition without considering self-adjoint matrices?

Thank yo uall in advance, Ziv Goldfeld

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    Are you sure you mean a "semi-ordering"? I don't think the second condition in [the definition of a semiorder](http://en.wikipedia.org/wiki/Semiorder#Definition) is fulfilled. Perhaps you mean a partial order?2012-11-26
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    I guess it's partial order. How do you compare [0,1;0,0] and [0,0;1,0]?2012-11-26

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