Consider an $n \times n$ positive semi-definite matrix $\mathbf{P}$. Consider a function $f(\mathbf{x})$ where $\mathbf{x}$ is a vector. Let the Jacobian of $f$ be defined as the $n \times m$ matrix $\mathbf{F} = \frac{\partial f}{\partial \mathbf{x}}$ where $\mathbf{F}$ is evaluated at some point $\mathbf{x}$.
If I left and right multiple $\mathbf{P}$ by $\mathbf{F}$, is the result positive semi-definite?
In other words, is $\mathbf{F}^T \mathbf{P} \mathbf{F}$ positive semi-definite?
More generally, is this true of any $n \times m$ matrix $\mathbf{F}$?
Best Answer
Hint: Take a good look at $x^TF^TPFx$. Knowing that $P$ is positive semidefinite, what can you say?