About trace of product of unitary and symmetric positive semidefinite matrices

Question

Suppose that $r,k,n$ are natural numbers such that $r\leq k\leq n$ and $X\in\mathbb{R}^{n\times n}$ is a symmetric positive semidefinite matrix. In addition, let $W\in\mathbb{R}^{n\times k}$ and $U\in\mathbb{R}^{n\times r}$ satisfy the following conditions: $$W^TW=I_k\qquad and\qquad U^TU=I_r,$$ where $I$ is the identity matrix. Is it possible to conclude the following relation? $$trace(U^TXU)\leq trace(W^TXW).$$ If the answer is negative, would you please give an counterexample.

Answer 1

This does not seem to be true in general. (I hope this example is not too trivial.)

Let $r=k=1$ and $n=2$. Let $U=\begin{bmatrix} 1\\0\end{bmatrix}$, $W=\begin{bmatrix} 0\\1\end{bmatrix}$, and $X=\begin{bmatrix} 1&0\\0&0\end{bmatrix}$.

Then $U^TU=W^TW=\begin{bmatrix}1\end{bmatrix}=I_1$, and $U^TXU=\begin{bmatrix}1\end{bmatrix}$ while $W^TXW=\begin{bmatrix}0\end{bmatrix}$, but $1\not\le 0$.