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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?

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New inner product has the form $(x,y)=\langle Bx,y\rangle$ for some positive definite (w.r.t $\langle\cdot,\cdot\rangle$) operator $B.$ So $(Tx,x)=\langle BTx,x\rangle\geq 0$ for all $x$ would mean that $BT$ is positive semidefinite w.r.t. $\langle\cdot,\cdot\rangle.$ This, of course, need not be the case.

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