This question is a generalisation of Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are rectangular matrices which itself is a generalisation of Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are square matrices.
Let $A$ be an $m \times n$ matrix and B and $n \times k$ matrix. Obviously, the matrix product $AB$ is possible, whereas the product $BA$ is not. Assume $n<k<m$, such that $AB$ is a large matrix.
Is there anything we can do to either matrix $A$ or $B$, such that the product $BA$ becomes possible and such that the eigenvalues of $BA$ say something about the eigenvalues of the original $AB$?
I am thinking of procedures such as:
- Truncating $A$ (making it $k \times n$)
- Appending some values to $B$ (making it $n \times m$)
- Interpolating values in $B$
- Taking random samples
Motivation 1 (theoretical): The matrix $AB$ is large and clearly degenerate. Therefore, there must be a smaller matrix which captures the same information as $AB$ (i.e. has the same eigenvalues). If $k=m$, then $BA$ would be such a smaller matrix, as discussed in Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are rectangular matrices.
Motivation 2 (practical): The eigendecomposition of a very large matrix is computationally expensive and may require special hardware. If the problem can be simplified, e.g. by decomposing the smaller $BA$, then the analysis can be performed more efficiently.
Alternatively, is there anything we can say about the eigenvalues of $AB$ without performing the product, i.e. based on analyses of $A$ and $B$ separately.