### Notation

• I use lowercase $x$ to represent scalars and vectors.
• I use uppercase $X$ to represent scalars and matrices.
• Unless otherwise stated, let $x$ be $N \times 1$ and complex.
• Unless otherwise stated, let $X$ be $M \times N$ and complex.
• $\sigma_{i}(A)$ is the $i$-th largest singular value of $A$.
• $\lambda_{i}(A)$ is the $i$-th largest eigenvalue of $A$.
• $\mathrm{vec}(A)$ is the column vectorization of $A$.

### Trace

(Cyclic Property) For properly sized matrices,