TITLE: Is trace invariant under cyclic permutation with rectangular matrices?
QUESTION [23 upvotes]: I'm working with trace of matrices. Trace is defined for square matrix and there are some useful rule, i.e. $\text{tr}(AB) = \text{tr}(BA)$, with $A$ and $B$ square, and more in general trace is invariant under cyclic permutation.
I was wondering if the formula $\text{tr}(AB) = \text{tr}(BA)$ holds even if $A$ and $B$ are rectangular, namely $A$ is $n$-by-$m$ and $B$ is $m$-by-$n$.
I figured out that if one completes the involved matrices to be square by adding zero entries in the right places, then the formula still works... but I want to be sure about this!

REPLY [37 votes]: Yes, it holds true. Let $A$ be a $n\times m$ and $B$ be a $m \times n$ matrix over the commutative ring $R$, we have
\begin{align*}
  \mathrm{tr}(AB) &= \sum_{i=1}^n (AB)_{ii}\\
    &=\sum_{i=1}^n \sum_{j=1}^m A_{ij}B_{ji}\\
    &= \sum_{j=1}^m \sum_{i=1}^n B_{ji}A_{ij}\\
    &= \sum_{j=1}^m (BA)_{jj}\\
    &= \mathrm{tr}(BA)
\end{align*}
So you can just prove the formula by computing.

REPLY [11 votes]: Yes, the cyclic invariance holds irrespective of the dimensions of the matrices. The trace of a product in either order is simply the sum of all products of corresponding entries.