What are the geometric multiplicities of the eigenvalues of a Kronecker (or tensor) product?

Question

What are the geometric multiplicities of the eigenvalues of a tensor (i.e. Kronecker) product of linear transformations? We can assume that we are working in finite-dimensional vector spaces over an algebraically closed field. $\DeclareMathOperator{\tr}{tr}$

I originally asked this question as part of this other question What are the algebraic multiplicities of the eigenvalues of a Kronecker (or tensor) product? I since decided to split it up. In the comment section of the linked post (credits to user3257842), an outline for an answer using Jordan normal forms became apparent. I will add this answer myself, but other perspectives are more than welcome! Especially abstract arguments :)

For reference, my first knee-jerk conjecture was the simplest possible, namely that $$ \gamma_{A\otimes B}(\lambda) = \sum_{\lambda_A\lambda_B=\lambda}\gamma_A(\lambda_A)\gamma_B(\lambda_B), $$

where $\gamma()$ denotes geometric multiplicity. This wasn't actually based on any examples , and turned out to be quite wrong. The true formula turns out to also depend on the generalized geometric multiplicities (i.e. on the particular Jordan block sizes).

Answer 1

Credits to user3257842 for the approach.

Let $A$ be a linear transformation of $V$ and $B$ of $W$. Choose bases $\{x_i\}_i$ and $\{y_i\}_i$ for $V$ and $W$ such that they both are in Jordan normal form. Then of course $\{x_i\otimes y_j\}_{i,j}$ is a basis for $V\otimes W$.

Write $A=\bigoplus_i J_{A,i}$ and $B=\bigoplus_j J_{B,j}$, where the $J$'s are Jordan blocks. Tensor products of linear maps distribute over direct sums, so $$ A\otimes B = \bigoplus_{i,j}J_{A,i}\otimes J_{B,j}. $$ (Why? We can order the basis $\{x_i\otimes y_j\}_{i,j}$ such that $A\otimes B$ is a diagonal block matrix of the desired form. Note that the lexicographical ordering doesn't give this structure, so we don't actually consider the Kronecker product of the matrices.)

Thus, the problem is reduced to studying the the properties of maps of the form $J_{\lambda, k}\otimes J_{\mu, l}$, where I've changed notation such that $J_{\lambda, k}$ is a Jordan block with eigenvalue $\mu$ and size $k$. We have $$ J_{\lambda, k}\otimes J_{\mu, l} - \lambda\mu I = \left(\begin{array}{ cccc | cccc | ccc | cccc } 0 & \lambda & & & \mu & 1 & & & & & & & & & \\ & 0 & \ddots & & & \mu & \ddots & & & & & & & & \\ & & \ddots & \lambda & & & \ddots & 1 & & & & & & & \\ & & & 0 & & & & \mu & & & & & & & \\ \hline & & & & 0 & \lambda & & & & & & & & & \\ & & & & & 0 & \ddots & & & \ddots & & & & & \\ & & & & & & \ddots & \lambda & & & & & & & \\ & & & & & & & 0 & & & & & & & \\ \hline & & & & & & & & & & & \mu & 1 & & \\ & & & & & & & & & \ddots & & & \mu & \ddots & \\ & & & & & & & & & & & & & \ddots & 1 \\ & & & & & & & & & & & & & & \mu \\ \hline & & & & & & & & & & & 0 & \lambda & & \\ & & & & & & & & & & & & 0 & \ddots & \\ & & & & & & & & & & & & & \ddots & \lambda \\ & & & & & & & & & & & & & & 0 \end{array}\right) $$ (I did my best on the formatting! ...). We are interested in the geometric multiplicity of the eigenvalue $\lambda\mu$, i.e. the nullity of $J_{\lambda, k}\otimes J_{\mu, l} - \lambda\mu I$.

From here on, I just reduce to row echelon form and count the non-zero rows. We have three cases:

• $\lambda=\mu=0$: The matrix is already in row echelon form with the $1$'s as the non-zero rows. We get a nullity of $k+l-1$.
• $\lambda=0, \mu\ne0$: The matrix is already in row echelon form with the $\mu$-rows as the non-zero rows. We get a nullity of $l$, i.e., in general, the size of the Jordan block with non-zero eigenvalue.
• $\lambda, \mu\ne0$. This time, the matrix is not yet in row echelon form. Assume WLOG that $k\le l$. Squinting at it a bit, the $\lambda$-rows can be chosen as the echelon rows, and the rest can be eliminated. Concretely, we can iteratively clean up row $l$, then rows $l$ and $2l$, and so on until rows $l,2l\ldots,(k-1)l$. This way, the rows $l,2l\ldots,(k-1)l$ will be eliminated, and it works as long as $k\le l$. In general, we get a nullity of $\min(k,l)$.

With this, we are done. The full formula becomes $$ \gamma_{A\otimes B}(\lambda) = \sum \gamma_{J_A\otimes J_B}(\lambda), $$ where $\gamma()$ denotes geometric multiplicity, and the sum is taken over Jordan blocks of $A$ and $B$ with eigenvalues whose product is $\lambda$. The terms are calculated according to the bullet points above.

I was surprised that the behaviour of the eigenvalue $0$ differs from the non-zero case like that!