# Questions tagged [kronecker-product]

The Kronecker product of two matrices $\mathbf{A}_{(K\times L)}=\{a_{kl}\}$ and $\mathbf{B}_{(M\times N)}=\{b_{mn}\}$ which is denoted by $\mathbf{A}\otimes\mathbf{B}$ is defined as $$\mathbf{A}\otimes\mathbf{B}=\mathbf{C}_{(KM\times LN)}=\begin{bmatrix}a_{11}\mathbf{B} &\dots & a_{1L}\mathbf{B}\\\vdots &\ddots&\vdots\\a_{K1}\mathbf{B} &\dots & a_{KL}\mathbf{B}\end{bmatrix}$$

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### A property of vec() operator and Kronecker product

I know the following relationship holds between the Kronecker product and the vectorization operator ($vec(.)$): $vec(ABC) = (C^T \otimes A)vec(B)$ A proof for this can be found here: https://www.ime....
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### Solving Kronecker product

I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance. Note: How I can derive $B(X \otimes Y)Q$ with respect to X and then ...
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### Hessian matrix of $\Lambda \mapsto y' (I + X\Lambda X')^{-1}y$

I have $$f=y'M^{-1}y$$ where $$M = I + X\Lambda X'$$ for $y \in \mathbb{R}^n$, $X\in \mathbb{R}^{n\times p}$, and $\Lambda$ is a $p\times p$ symmetric positive-definite matrix. I'm trying to compute ...
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### What are the geometric multiplicities of the eigenvalues of a Kronecker (or tensor) product?

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 ...
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### What are the algebraic multiplicities of the eigenvalues of a Kronecker (or tensor) product?

What are the algebraic 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 ...
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### Kronecker product between matrices

I have to proove that $(K_{p,p}⊗ I_p)(\vec{a}⊗\vec{a}⊗\vec{a})$ equals $(\vec{a}⊗\vec{a}⊗\vec{a})$, where $\vec{a}$ is px1 vector, $I_p$ is unit matrix pxp and $K_{p,p}$ is commutation matrix ppxpp. I ...
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### Why does the Kronecker (tensor) product appear in this system of equations involving commutators of linear transformations?

Main question: Is there any abstract explanation of why the tensor (Kronecker) product of maps turns up in the situation given below? I was working on the same question from Halmos as this one, ...
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### How to write summation as matrix form

I am new to Matrix. Let $(\tau_{nij})_{N\times N\times J}$ $(X_{nj})_{N\times J}$, $(\gamma_{n jk})_{N\times J\times J}$, $(\pi_{i n j})_{N\times N\times J}$ and $(\alpha_{nj})_{N \times J}$, $(D_n)_N$...
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### A Hierarchical matrix block concatenation and kronecker products. Can I ensure representability?

I have been thinking of hierarchical block matrix products and how they can relate to a special class of sums of Kronecker products. Let us say I have a dictionary of matrices $\{M_{1},\cdots,M_N\}$ ...
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### On kronecker product representation of $AA^t$

Proposition: Prove that orthogonal matrices $\mathrm O_n(\mathbb{R})$ is a regular submanifold of $\operatorname{GL}_n(\mathbb{R})$. Proof: Denote \$f:\operatorname{GL}_n(\mathbb{R})\to \operatorname{...