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<m$ .

  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?