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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28 views

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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Weird linear matrix equality

I'm trying to understand how to guarantee the existence of a solution pair $(C,d)$, where $C \in \mathbb{R}^{n \times m}$ and $d \in \mathbb{R}^{n}$, to the following (weird) matrix equality: $$ (C \...
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What is the Kronecker Product of two vectors?

In my numerical methods course we got a homework problem that has a definition of a function $\phi(x) = vec(M) - x \otimes x $ where $x\otimes x$ is the kronecker product of an n-vector and $ M $ is ...
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38 views

Identity with traces and a Kronecker product

I'm studying on Matrix Variate distributions by Gupta and Nagar. I can't figure out why the following identity holds: \begin{equation}\text{tr}\{(\Sigma^{-1}\otimes \Psi^{-1})(\boldsymbol{x}-\...
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Kronecker factorization

I have found in a paper the following identities (equations $(119)$, $(120)$, appendix D): \begin{equation}\begin{aligned} \sum_{i,j=1}^s p_{ij} d_i d_j'X &= \begin{bmatrix}d_1 \\ \vdots \\ d_s\...
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Compute exponential of sum of Kronecker products of non-commuting matrices

In the context of CTMCs I want to compute following matrix exponential: $$\exp(M_1\otimes M_2 + M_3\otimes M_4 + D_1\otimes D_2 + D_3\otimes D_4)$$ with: $\otimes$ being the Kronecker product, all ...
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Find a rotation matrix with two constraints: Aligns two vectors AND a third vector is perpendicular to a fourth vector when transformed

I have four vectors $ d,w,c,n \in \mathscr{R}^3$. I want to find a rotation matrix $R$ that satisfies these constraints: $w$ is aligned with $d$ after rotation AND $c$ is perpendicular to $n$ after ...
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express a matrix using Kronecker product

How to express the following matrix as a Kronecker product of two $2\times 2$ matrices: \begin{equation*} A = \begin{pmatrix} 0 & 0 & 0 & 1+j \\ 0 & 0 & 1-j& 0 \\ 0 & 1-j &...
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28 views

Jacobian of F(X)=XBX involving kronecker product.

I'm trying to prove the following: Let $X_{m\times m}$ be a symmetric matrix of variables. Find the Jacobian, defined as: $$\frac{\partial}{\partial vec{X'}}vec{F}$$ where $F(X)=XBX$ , where $B_{m\...
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Kronecker product of two projection matrices

Let $M$ and $N$ denote subspaces of a finite-dimensional Euclidian space such that $N \subset M$. Let $P_M$ and $P_N$ be the projection matrix onto the subspaces $M$ and $N$ respectively. We know that ...
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52 views

Determinant of Kronecker Product

Problem. Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. Show that $$\det(A\otimes B)=\det(B\otimes A).$$ This formula apparently holds if $A$ and $B$ are square matrices, but here ...
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How to construct Jordan normal form for Kronecker product given known Jordan form for factors?

I am aware of simple formulas deriving singular and eigen values of kronecker product of matrices from their respective singular and eigen values. Does there similarly exist some generalization for ...
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42 views

Nearest Kronecker Product of a commutation matrix

Given a commutation matrix $K$ of the size $n²$ by $n²$. How can I find matrix $A$ and $B$ both of the size $n$ by $n$ such that the equation below is (approximately) satisfied by finding its Nearest ...
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How to write 3rd-or-higher order polynomials for matrices?

The other week I built a couple of line search algorithms for step-wise optimization on manifolds using gradients and Hessians. Up to second order pose little problem as this can be readily described ...
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tensor product and matrix multiplication distributive properties

I am trying to find partial trace of some matrix of the form $M = (A \otimes B)\times (A^{T*} \otimes B^{T*})$ in which $\otimes$ is tensor product, $\times$ is matrix multiplication, $T*$ is ...
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How to multiply a vector and a square matrix with Kronecker product, and know the answer's shape? [closed]

$\mathbf{1}_n \in \mathbb{I}^{n\times 1}$ is a vector of ones with shape $n\times 1$ $\mathbf{I}_m \in \mathbb{I}^{m\times m}$ is an identity matrix with shape $m\times m$ What is the answer to, and ...
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Mixed Product Property: Kronecker Hadamard Matrix Multiplication

Given matrices $A,B, C, D$, I wonder whether there is a way to simplify $$(I_d\otimes A)(B\odot C)(1_d\otimes D),$$ where $A$ is $n\times m$, $B, C$ are $dm\times dn$, and $D$ is $n\times k$. Moreover,...
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37 views

Nuclear norm distance under Kronecker product

Let $A\otimes A$ denote the Kronecker product. Suppose $\|A - B\|_1 = \varepsilon$, where $\|\cdot \|_1$ is the nuclear norm a defined by $\|X\|_1 = \text{Tr}(\sqrt{X^\dagger X})$ and $X^\dagger$ is ...
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Derivative including kronecker product

Suppose that $\delta$ is a $p\times1$ vector and $M$ is a symmetric $Np\times Np$ matrix. I'm trying to differentiate $\ln L=\ln|(I_{N}\otimes\delta')M(I_{N}\otimes\delta)|$ with respect to $\delta$ , ...
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How to efficiently solve a least squares problem involving Kronecker product and Tikhonov regularization

I have the following regularized least squares problem: $$ \min_x \|y - Ax\|_2^2 + \lambda \|Dx\|_2^2, $$ where $y \in \mathbb{R}^m$, $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, $D$ is a ...
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127 views

Product of Two Kronecker delta

Assume that the range of dummy indices is from 1 to N $$\delta_{ij} \delta_{jn} = \delta_{i1} \delta_{1n} + \delta_{i2} \delta_{2n} + \delta_{i3} \delta_{3n} +\cdots + \delta_{ii} \delta_{in} + \cdots ...
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34 views

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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3answers
74 views

Differentiate with Kronecker product

Acutually, I have a function that : $$\operatorname{tr}(\mathbf{M}(\mathbf{B}\otimes\mathbf{A}))$$ where $M$ and $B$ are constant matrix while $A$ is my variable. I want to have this : $$d \...
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1answer
64 views

Matrix corresponding to transformation $T(A\otimes B)=B\otimes A$

Is there a name for the matrix which corresponds to commuting the the order of Kronecker product? I'm interested in relating it to other named matrices like the commutation matrix. More precisely, ...
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1answer
53 views

Eigenvalues and matrix kronecker product

I'm not able to understand why this equivalences are true for the kronecker product of a matrix and why a the eigenvalues of a kronecker product of two matrixes are the product of their eigenvalues. ...
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77 views

Reducing the Kronecker Delta

I am not sure where my line of thinking for $\delta_{ij} \delta_{ij} = \delta_{ii}$ is going wrong. Please help me find my error:
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43 views

Eigenvalues of the sum of a Kronecker product (of two diagonal matrices) with a Kronecker sum [A⊕B+D1⨂D2=A⨂I+I⨂B=D1⨂D2]

We want to find the eigenvalues of the matrix $M \in \mathbb{R}^{n^2 \times n^2}$, which is the sum of a Kronecker sum and a Kronecker product, that is $$ M = A \oplus B + D_1 \otimes D_2 = A \otimes ...
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138 views

Can we obtain min $\mathbf{x}$ in $\Vert A - B(I\otimes \mathbf{x})\Vert_F^2$ algebraically?

Are we able to obtain the following algebraically? $$ \widehat{\mathbf{x}}= \underset{\mathbf{x}}{\operatorname{argmin}}\Vert A - B(I\otimes \mathbf{x})\Vert_F^2 $$ where $A\in\mathbb{R}^{m\times n},\,...
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Evaluating a kronecker delta product

Say I have a product, $\sum_{n=0}^{+\infty} \sum_{k=0}^{+\infty}q_{n}r_{k}\delta_{n-1,k-1}$. How do I evaluate this? I've tried opening up the $n$ summation, which gives me, $\sum_{k=0}^{+\infty} \...
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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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1answer
70 views

How to rewrite a matrix expression involving Kronecker product and trace as a quadratic form?

For a vector $\mathbf{x} \in \mathbb{R}^n$ and symmetric matrices $\mathbf{A} \in \mathbb{R}^{n\times n}$, $\mathbf{B} \in \mathbb{R}^{n^2\times n^2}$, I want to find an expression for the matrix $\...
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1answer
38 views

I don't understand the fast generation algorithm for Stochastic Kronecker Graphs

I've been reading this paper and using this link as a reference while reading about Stochastic Kronecker Graphs, and I don't understand the algorithm that generates such a graph recursively and in $O(...
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1answer
84 views

Rewriting summation over vector-matrix-vector products as one vector-matrix-vector product?

I have an expression for a scalar, which is the double sum over vector-matrix-vector products. All terms are real-values vectors/matrices and dimensions are below each term: $$\sum_{i=1}^N \sum_{j=1}^...
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75 views

Gradient of Kronecker product

I have these expressions, which are the gradient of a number of constraints on an optimization problem $$\frac{\partial c}{\partial z_r}=B(I\otimes z_r + z_r\otimes I - iI\otimes z_i + iz_i\otimes I) \...
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36 views

Positive definite of integral kronecker product of two symmetric functions

Let $\Gamma_{\min}$ and $\Gamma_{\max}$ denote the smallest and largest eigenvalues of a matrix. $\otimes$ is the Kronecker product. Given bounded continuous matrix function $A(u_1,u_2)$ and $B(u_1,...
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First matrix is a Kronecker product. Second is not. Can it be wriyten as a Kronecker product?

Can somebody please tell how can I write the matrix system [ b11 b12 0 0; b21 b22 0 0; 0 0 b11 b12; 0 0 b21 b22][u11 u21 u12 u22] into [b11 b12 0 0; 0 0 b11 b12; b21 b22 0 0; 0 0 b21 b22][u11 u12 u21 ...
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1answer
18 views

Orthonormal columns of block matrices expanded with Kronecker products

Let $W_{i,1},W_{i,2},W_{i,3} \in \mathbb{R}^{n \times n}$, $i \in {1,2}$ be such that $$ \eqalign{ \Big[\matrix{W_{i,1}^T & W_{i,2}^T & W_{i,3}^T}\Big] \left[\matrix{W_{i,1}\\W_{i,2}\...
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2answers
44 views

Derivative of a trace with relation to a vector inside a kronecker product

I'm trying to obtain the derivative wrt $\beta$ in $\textrm{Tr}(A(I_n \otimes \beta)B(I_n \otimes \beta))$. I've tried to follow the same procedure as this question Derivative of a trace with second ...
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1answer
44 views

Block matrix of tensor product

$K$ is a field, $K^n$ is a vector space with $(e_1, \ldots, e_n)$. The tensor product $K^n \otimes K^n$ has the basis $\mathcal {B} = (e_1 \otimes e_1, \ldots, e_1 \otimes e_n, e_2 \otimes e_1,, \...
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1answer
80 views

Gradient of matrix operations

Assume we have a matrix $M\in \mathbb R^{t\times qt}$, a vector $p \in \mathbb R^{r}$, and a vector $z \in \mathbb R^{qt}$, . Note that $\otimes$ is a Kronecker product and $\odot$ is a Hadamard ...
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1answer
80 views

Cholesky decomposition of a Kronecker product

Assume that the $n\times n$ matrix $\mathbf{A}$ has the Cholesky decomposition of the form $\mathbf{A}=\mathbf{L}\mathbf{L}^H$. Now, suppose the matrix $\mathbf{B}$ is the result of a Kronecker ...
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1answer
46 views

Calculate gradient of Kronecker product.

Assume we have a matrix $M\in \mathbb R^{m\times n}$ and two vectors $z\in \mathbb R^{n}$ and $p\in\mathbb R^{k}$. Then, define $\mathbb 1_{i\times j}$ is $i$-by-$j$ vector whose elements are all 1. $\...
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2answers
56 views

Transformation of mixed Kronecker and Hadamard products

Given $ w = \Big((A \otimes 1)\odot(1^T \otimes x)\Big)\ d $, express $ w $ in the form $ w = Q x $, where $ Q, A $ are matrices and $ d $ is a vector. I have tried to solve this by using the mixed ...
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51 views

tensor of operators is Kronecker product?

Say $A$ and $B$ are operators on Hilbert spaces $H_A,H_B$ respectively. If the Hilbert spaces are finite dimensional, then I know the tensor $A\otimes B$ can be represented by the Kronecker product $[...
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1answer
149 views

Frobenius norm involving Kronecker Product

Consider $ J = ||\mathbf{G} - ( \mathbf{B} \otimes \mathbf{X} )||_F^2 $, where $\mathbf{G}$ and $\mathbf{B}$ are complex matrices, and $||.||_F$ is the Frobenius norm. Find the derivative with respect ...
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1answer
23 views

What is the name of the operation when you do a Kronecker tensor square without repeated combination of products?

let's say I have a tensor: A = [3, 5, 7] If I do a Kronecker product of A with A I would get ...
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1answer
135 views

Derivative of a trace with second order Kronecker product

I am trying to compute the derivative of $J$ with respect to $F$. when $$ J = \mathrm{Tr}\lbrack(I_{N} \otimes F)^{T}A(I_{N} \otimes F)B\rbrack $$ $$ F \in \mathbb{R}^{N \times Nn},\ \ A \in \mathbb{...

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