Hermitian scalar product: When is it invertible?

Question

I know that positive definite symmetric matrices (in euclidean vector spaces) are always invertible. Are positive definite symmetric hermitian matrices always invertible (in an hermitian vector space) as well? I don't know how to be sure and I can't prove it. Help?

I'm supposing $V$ is an hermitian vector space, $v$ is a vector in $V$ and $\langle \cdot , \cdot \rangle$ hermitian scalar product. Given any $v$ in $V$, $\langle v, v \rangle$ is real, so my definition of "positive definite" is:

$$ \langle v, v \rangle > 0$$

for all non-zero $v$ in $V$ and

$$ \langle v, v \rangle = 0$$

iff $v=0$.

Answer 1

Your definition should say $\langle v,\,v\rangle\ge0$ for $v\in V$, with equality iff $v=0$. To prove invertibility, note the matrix's eigenvalues must be positive.