TITLE: Why is positive (semi-)definite only defined for symmetric matrices?
QUESTION [24 upvotes]: When we are defining positive (semi-)definite matrices, we do so for symmetric matrices only.
Why do we need symmetry in the definition?

REPLY [25 votes]: It's actually defined more generally for Hermitian matrices (matrices equal to the complex conjugate of their transpose).  If a Hermitian matrix has at least one entry with a (nonzero) imaginary part, then the matrix is not symmetric, but it could still be positive (semi-)definite.
There are two reasons I can think of why we wouldn't extend the definitions to non-Hermitian matrices.  The first is that if the matrix is not Hermitian, then we could have complex eigenvalues.  In that context, "positive" has no meaning.
Another reason why I think we wouldn't has to do with quadratic forms.  If $A$ is a real symmetric matrix, and $x$ is a vector of variables, then $f(x)=x^TAx$ is called a quadratic form.  $f(x)>0$ ($f(x)\geq 0$) for all nonzero $x$ if and only if $A$ is positive definite (semi-definite).  We could relax the condition that $A$ is symmetric, and define quadratic forms in the same way for a general real matrix.  But the resulting quadratic form is the same as the one defined by the real symmetric matrix $(A+A^T)/2$.  The quadratic form is positive definite if and only if $(A+A^T)/2$ is, so positive definiteness is not a property of $A$ as much as it is a property $(A+A^T)/2.$  In other words, we would end up studying symmetric matrices anyway, so wouldn't get anything new by extending the definition to non-symmetric real matrices.