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In some definitions of "definite" for matrices, it is required that it should be "symmetric". Why is that?

When you use the definition

$M \text{ positive definite} \Leftrightarrow \forall x : xMx^T > 0$

I don't see, why $M$ would have to be symmetric for that to make sense and to be written down.

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    If we do not demand symmetric, note that $M$ is positive definite iff the symmetric matrix $\frac 12(M+M^T)$ is. In fact, $xMx^T=x(\frac12(M+M^T))x^T$. – 2017-01-25
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    the short answer is because one usually likes to define bilinear forms for such matrices, which should be symmetric in their arguments. – 2017-01-25

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