Does changing the inner product preserve positive-definite automorphisms?

Does changing the inner product preserve positive-definite automorphisms?

Let $V$ be a finite-dimensional complex vector space equipped with an
inner product $\langle\bullet,\bullet\rangle$. Let $T$ be an automorphism
of $V$. Suppose that $T$ is positive-definite relative to
$\langle\bullet,\bullet\rangle$, that is, $\langle v, Tv\rangle >0$ for
all $v$ in $V$.
Given another inner product $(\bullet,\bullet)$ on $V$, is $T$
positive-definite relative to this new inner product?