Basic Properties of the Trace

Let $p,q\ge1$ be natural numbers. Products below are matrix products, $(\cdot)^{\top}$ is the transpose, and $\operatorname{tr}$ is the trace.

1. (Linearity) For real $p\times p$ matrices $M,N$ and real numbers $a,b$, where $aM+bN$ denotes the entrywise linear combination: $\operatorname{tr}(aM+bN)=a\operatorname{tr}(M)+b\operatorname{tr}(N)$.

2. (Transpose invariance) For every real $p\times p$ matrix $M$: $\operatorname{tr}(M^{\top})=\operatorname{tr}(M)$.

3. (Cyclic property) For a real $p\times q$ matrix $U$ and a real $q\times p$ matrix $V$: $\operatorname{tr}(UV)=\operatorname{tr}(VU)$.

4. (Entrywise sums) For real $p\times q$ matrices $U$ and $V$: