## Matrix inversion of the Kronecker product of matrices

How can I compute the inverse of $$U=\lambda\mathbb{I}-(S\otimes S)^T(S\otimes S)$$ matrix, where $$S$$ is an $$N\times M$$ matrix using something like rank-one update?

I must remove the $$i$$-th row of $$S$$ in a loop over its rows and recompute the inverse of $$U$$. It would be computationally very expensive since $$N\ggg1$$. Can anybody suggest a way to just with some kind of rank-one update I will be able to compute $$U^{-1}$$? For instance using some kind of a Sherman-Morrison-Woodbury formula, similar to this paper in equations (51) to (54).

## Pseudo-inverse of “sandwiched” Kronecker product

Let $$\otimes$$ denote the Kronecker product. I know that for two matrices $$A$$ and $$B$$, $$(A \otimes B)^\dagger = A^\dagger \otimes B^\dagger =: \Omega^\dagger$$, where the $$\dagger$$-superscript denotes the Moore-Penrose inverse. Now let $$E$$ be a diagonal 0-1 matrix (arbitrary positions) conformable with $$\Omega$$. Hence, $$E$$ is an orthogonal projection. Given $$A$$ and $$B$$ are invertible, what can we say about $$(E \Omega E)^\dagger?$$

It seems tempting to somehow exploit the property $$E^\dagger = E$$. Yet clearly, we have $$(E \Omega E)^\dagger \neq(E\Omega^\dagger E)$$. So far, I could not find something useful on this case.

In case nothing can be said under this generality, is any combination of the following additional qualifications useful:

• $$A$$ is symmetric and circulant
• $$A$$ equals $$I – \gamma G_c$$, where $$I$$ is the identity matrix, $$\gamma > 0$$ is some parameter, and $$G_c$$ is the adjacency matrix of the complete graph
• $$B$$ is symmetric and positive definite
• $$B$$ is eventually positive

“Eventual positivity” is defined as $$\exists k \in \mathcal{N}: B^l \geq 0 \, \forall\, l \geq k$$, where the matrix inequality is entrywise.

## Rewrite kronecker product of identity plus something

I’m working on trying to find a way to get the eigen-values of a complicated matrix but all the original elements themselves are either block-diagonal (as in, all blocks are the same also) or some simple repeated matrix. Computationally, I see a pattern of repeated eigen-values that makes me think I could write them in a more analytical form. I’m however getting into issues because there’s a $$I_n – D$$ like term that’s making it troublesome for me. This question is a smaller piece from this though.

Just to be clear on some notation. Let $$J_k = 1_k1_k’$$ be the square matrix of size $$k$$ of all 1s. Let $$A$$ and $$B$$ be square matrices of size $$c$$; $$A,B\in\mathbb{R}^{c \times c}$$.

Is there anyway to rewrite the following sum of kronecker products in such a way that find eigen-values would be “simple”?

$$\left( I_k \otimes A \right) + \left(J_k \otimes B \right)$$

where all matrix multiplication is possible. I think if it’s possible it has something to do with a clever way of making $$I_k$$ and $$J_k$$ look more similar to each other but I haven’t been able to think of anything.

Specifically in my case $$A$$ and $$B$$ have some similar structure in that $$A=Z’WZ$$ and $$B=Z’WCZ$$ but any points on the more general problem may be helpful.

There is a “smaller” piece earlier of the form $$I_{kc} – J_k \otimes D$$, $$D\in\mathbb{R}^{c\times c}$$, that if I had some other way to rewrite might make the later things simpler.

## Kronecker Product of Multiple Matrices

I would like to construct the following matrix: $$Z\otimes I\otimes I\otimes … \otimes I\+ I\otimes Z\otimes I \otimes …\otimes I\+ …\+Z\otimes I\otimes I\otimes … \otimes Z$$ where $$Z$$ is Pauli Z matrix, and I is 2 dimensional identity matrix, and in each term, there are $$N$$ such 2dimensional matrices kronecker product together.

In mathematica, How can I write a code with input $$N$$ to construct such matrix? For example, when $$N=3$$, we have

KroneckerProduct[PauliMatrix[3], PauliMatrix[0], PauliMatrix[0]] + KroneckerProduct[PauliMatrix[0], PauliMatrix[3], PauliMatrix[0]] + KroneckerProduct[PauliMatrix[0], PauliMatrix[0], PauliMatrix[3]] 

But I am not sure how to do this in a versatile way, where we only need to input the parameter $$N$$ and do not have to rewrite the code again for each $$N$$.

Thanks!

## Rank of certain ‘Kronecker Column Hadamard Row Sum’ of matrices

Define Kronecker Column Hadamard Row Sum of two matrices $$M_1$$ and $$M_2$$ of size $$n_1\times m$$ and $$n_2\times m$$ respectively to be the $$n_1n_2\times m$$ matrix whose $$((i-1)n_1+(j-1))$$th row is sum of $$i$$th row of $$M_1$$ and $$j$$th row of $$M_2$$.

Take $$n$$ different matrices $$M_1,\dots,M_n$$ with ranks $$r_1$$ through $$r_n$$ respectively and sizes $$m_1\times m$$ through $$m_n\times m$$ respectively and with $$\prod_{i=1}^n m_i.

1. How large can the rank of Kronecker Column Hadamard Row Sum of $$M_1,\dots,M_n$$ be?

2. If each of $$M_1,\dots,M_n$$ is picked uniformly from matrices with ranks $$r_1$$ through $$r_n$$ respectively and sizes $$m_1\times m$$ through $$m_n\times m$$ respectively then what is the expected rank of Kronecker Row Direct Sum of $$M_1,\dots,M_n$$ and would an inequality like Chebyshev’s expected to hold?