Let $\bf P$ be a $n \times n$ symmetric positive definite matrix. It is known that
$$ \operatorname{tr} ({\bf P}) \geq n + \log \det({\bf P})$$
Since ${\bf P} \succ {\bf O}_n$, it is also known that $\bf P$ is invertible. Hence,
$$ \operatorname{tr} \left( {\bf P}^{-1} \right) \geq n + \log \det \left( {\bf P}^{-1} \right) = n - \log \det({\bf P}) $$
Adding these two inequalities,
$$ \color{blue}{\operatorname{tr} ({\bf P}) + \operatorname{tr} \left( {\bf P}^{-1} \right) \geq 2 n} $$
Has this trace inequality earned a name?
Alternatively, this trace inequality can be derived from the fact that $x + \frac{1}{x} \geq 2$ for all $x > 0$.
Related
- Show that $A+A^{-1}\geq 2I$ [hat tip: Dietrich Burde]
