$|a^Tb| \leq \|a\|_{M} \|b\|_{M^{-1}}$ where $M$ is positive definite matrix and $a, b \in \mathbb{R}^d$

Question

Is the following inequality true? If so what is it called?

Let $M \in \mathbb{R}^{d\times d}$ be a positive definite matrix and $a, b \in \mathbb{R}^d$. Then

\begin{equation} |a^Tb| \leq \|a\|_{M} \|b\|_{M^{-1}} \end{equation}

where $\|a\|_{M} = \sqrt{a^T Ma}$

It does look like Holder inequality but I am not sure if Holder holds for matrix induced norm as well