# 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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### In the case where ⊕ is involved as a composite system M1 ⊕ M2, what kind of equation does ⊕ denote?

I am reading about relations between equations using kronecker products as well as the use of ⊕ in situations such as M1 ⊕ M2, M ⊕ N, S ⊕ S, S⊕S⊕…n and others besides. In these papers they are called ...
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### An eigen-decomposition problem with Kronecker product

First define a function $P$ mapping matrix into matrix of dimension $m$ by $m$, given $V$ and $L$ $$P(A)=V^T(L\otimes A)(L\otimes A)^TV$$ where $L$ is $N$ by $N$ and $V = [v_1,v_2,\ldots,v_m]$, {$v_i$}...
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### Kronecker “square root” of a definite positive matrix

Let $\Sigma$ be a $m\cdot n\times m\cdot n$ symmetric and positively definite matrix. I wonder if someone know under what hypotheses one can find (and how) a decomposition of the type \begin{equation}\...
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### Spectra and norm of Kronecker product

I read a statement in a paper and I cannot understand why it is true. Let $A,B$ be symmetric real matrices of possibly different sizes, with eigenvalues $(\lambda_k)_k$ and $(\gamma_j)_j$. Then it is ...
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### Fourier transform of a Kronecker product

Let $v=v(\theta)$ and $w=w(\theta)$ vectors with $d$ components defined as \begin{align} v&=(e^{i\theta}\quad e^{i2\theta}\quad \ldots\quad e^{id\theta})\\ w&=(e^{i\theta(\rho_1-1)}\quad e^{i\...
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### Positive semidefiniteness of operator containing Kronecker product, commutation and orthogonal matrices

I have $\mathbf{G} \in \mathbb{R}^{k^2 \times k^2}$ be defined by: \begin{equation} \mathbf{G} = (\mathbf{K}_{k^2}+\mathbf{I}_{k^2}) \left(\mathbf{I}_k \otimes (\mathbf{X})^T \right) \mathbf{J} (\...
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