On the similarity of matrices $AB$ and $BA$ over a field

Let $A$ and $B$ be $n$-by-$n$ matrices over a field. The study of the relationship between the products of matrices $AB$ and $BA$ has a long history. It is well-known that $AB$ and $BA$ have equal characteristic polynomials (and, therefore, eigenvalues, traces, etc.). One beautiful result was obtained by H. Flanders in 1951. He determined the relationship between the elementary divisors of $AB$ and $BA$, which can be seen as a criterion when two matrices $C$ and $D$ can be realized as $C = AB$ and $D = BA$. If one of the matrices ($A$ or $B$) is invertible, then the matrices $AB$ and $BA$ are similar. If both $A$ and $B$ are singular then matrices $AB$ and $BA$ are not always similar. We give conditions under which matrices $AB$ and $BA$ are similar. The rank of matrices plays an important role in this investigation.

Gauge invariant information concerning quantum channels

Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.