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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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Kronecker Product Interpretation

The algebraic expression for a Kronecker product is simple enough. Is there some way to understand what this product is? The expression for matrix-vector multiplication is easy enough to understand. ...
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kronecker product to find a matrix

what is hk? I declared my question in image kronecker product to form a matrix i want to use this as a part of my matlab code, tnx!
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How to rewrite kronecker products as linear combination of matrices

Is it possible to write a matrix $U$ such that $U \otimes U \approx A\Lambda A^T$, and $U \approx AB\Lambda_1B^TA^T$? Here $U \in R^{n \times n}$, $A \in R^{n \times k}$, $B \in R^{k \times k_1}$, and ...
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Differentiation of tensor product

I have a tensor equation $$\frac{\partial A_{ij}}{\partial B_{kl}}=\frac{\partial A_{ij}}{\partial C_{pq}}\frac{\partial C_{pq}}{\partial B_{kl}} $$ $C_{pq}$ can be written as $C_{pq}=B_{pq}+aB_{mm}\...
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A Kronecker Product identity

I want to show the Kronecker Product identity listed on Wikipedia: $$\begin{align} \mathrm{vec}(AXB) =(B^T \otimes A) \mathrm{vec}(X) \\ \tag{1} \end{align}$$ Wikipedia does not cite references for ...
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How to prove eigenvalues of specific block matrix are as proposed

In some of my work (statistics), I need the eigenvalues of a very large matrix. As such I would like to reduce it to a simpler problem and it seems entirely possible to me as the matrix has a very ...
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1answer
42 views

Proving matrix equality $(K_{n,n}\otimes I_n)a^{\otimes3}=a^{\otimes3}$

How to prove the matrix equality $(K_{n,n}\otimes I_n)a^{\otimes3}=a^{\otimes3}$? Here $K_{n,n}$ is a $n^2\times n^2$ commutation matrix, $I_n$ is a $n\times n$ identity matrix and $a$ is a $n\times1$...
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1answer
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Permute the factors of a Kronecker product

Let two matrices $A$ and $B$ of size $m\times n$ and $p \times q$, respectively. What is the expression of two matrices $F$ and $G$ such that $A \otimes B = F ( B \otimes A ) G$?
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Matrix expression for $\operatorname{vec}(X^{\top}X)$?

Let $X$ be an $m$ by $n$ matrix. I would like to find the $n^2$ by $(mn)^2$ matrix $B$ such that $$ \operatorname{vec}\left(X^{\top}X\right) = B \operatorname{vec}\left(\operatorname{vec}\left(X\right)...
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integral of a Kronecker product of exponentials

Let $A\in\mathbb{R}^{n\times n}$. I do not know how to get a solution to the following integral: $\int_{0}^{t}\left( e^{As}\otimes e^{As}\right) ds$
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How to efficiently compute the matrix-vector product $y = (I_p \otimes A \otimes I_r )x$

Is there a way to compute $y = (I_p \otimes A \otimes I_r )x$ efficiently? Where $A \in {\rm I\!R}^{q\times q}$ and $x ∈ {\rm I\!R}^{pqr}$? I know that for $y = (I_p\otimes A)x$ this can be written ...
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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 ...
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2answers
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Basic property of kronecker delta $( A \otimes B)(A^{-1} \otimes B^{-1})$

Given non singular matrices $A_{n \times n},B_{m \times m}$ $$ ( A \otimes B)(A^{-1} \otimes B^{-1}) = (AA^{-1}) \otimes (BB^{-1}) = I_n \otimes I_m = I_{(nm \times nm )} $$ I was just reading ...
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Why is the Kronecker sum defined for square matrices?

Background From Wikipedia, if A is an $m\times n$ matrix and B is a $p\times q$ matrix, the the Kronecker product $\mathbf A\otimes \mathbf B$ is the $mn\times nq$ block matrix $$\mathbf A\otimes \...
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Factoring a matrix as the product of block triangular and diagonal matrices.

How can I check that the matrix $$K = \left[\begin{array}{c|cc} 1 & 0^{\mathrm T}_m & m \mathbf u^{\mathrm T} \\ \hline 0_m & I_m & I_m \\ m \mathbf u & I_m & O_m \end{array}...
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Writing a matrix in an alternative form with a Kronecker product.

I need to express the matrix \begin{equation} \begin{bmatrix} I & A \\ A^T & O \\ \end{bmatrix} \end{equation} where $$A = \begin{bmatrix} m\textbf{u}^T\\ I_m\\ \...
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1answer
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Derivative with respect to vectorized inverse Kronecker product

I am trying to derive the gradient of a function I wish to optimize, and wish to obtain the following derivative: $$ \frac{\partial}{\partial \pmb{x}} \left(\pmb{I} - \pmb{X} \otimes \pmb{X} \right)^{-...
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1answer
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Feature matrix as the Kronecker product of two feature matrices. How to build an alternative?

I have two feature matrices $\textbf{X}$ and $\textbf{Y}$ which I encoded through one-hot encoding the rows of two feature matrices $\textbf{X'}$ and $\textbf{Y'}$. Thus, they are sparse with a few $1$...
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1answer
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Matrix equation of Kronecker product

$\ I ⊗ I ⊗ A = Z ⊗ I$ $A$ is a known matrix. $I$ is the identity matrix. $A$ and $I$ are n by n matrices. $⊗$ is the Kronecker product. Is there a way to find $Z$ in terms of $A$ and $I$? Thanks in ...
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55 views

Derivative of vectorized kronecker product

I'm struggling with the following derivative. Let $\pmb{X}$ be a symmetrical $n \times n$ matrix, $\pmb{x} = \mathrm{vec}(\pmb{X})$ and let function $\pmb{f}$ take the following form: $$ \pmb{f} = \...
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1answer
39 views

Derivative with respect to vector of product of two functions of the vector

I am struggling with the following derivative. Let $\pmb{x} \in \mathbb{R}^{n}$ be a vector, $\pmb{y} \in \mathbb{R}^{m}$ another vector that is a function of $\pmb{x}$, and $\pmb{g}$ and $\pmb{h}$ ...
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Derivative of Kronecker product of vector with itself

I'm struggling with the following problem. Suppose $\pmb{x}$ and $\pmb{y}$ are vectors of the same length and $\pmb{y}$ is not a function of $\pmb{x}$. What is the following derivative? $$ \frac{\...
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1answer
58 views

Kronecker product of identity and matrix product

How is the following property true? Let $I$ be the identity matrix and $A$, $B$ be appropriately sized real matrices. Then $$I \otimes \left(\left( I \otimes A\right) B \right) = \left( I \otimes I \...
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Property on Kronecker product

I read a paper and there was an equation which was finally derived an equivalent expression as $$ L = L_{T} \otimes I_{G} + I_{T} \otimes L_{G} = {\color{blue}{L_{T} \times L_{G}}} , $$ and ...
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1answer
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How to rewrite $\left[\begin{smallmatrix} A\otimes B_1\\ \vdots\\ A\otimes B_T \end{smallmatrix}\right]$?

We have the following matrix $\left[\begin{array}{c} A\otimes B_1\\ \vdots\\ A\otimes B_T \end{array}\right]$, where $A$ and the $B_i$ are matrices, and $\otimes$ is the Kronecker product. Is it ...
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Question about creating a 3-rank tensor in numpy with python3

I'm trying to create a 3-rank tensor in numpy, with python3.x. I need to create this 3-rank tensor A in a very particular way. If I have 3 matrices, let's say that all of them are some $Y$ matrix (...
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1answer
105 views

Matrix Differentiation of Kronecker Product

I have a question about differentiating an expression which has multiple kronecker products. I have the following objective function I would like to differentiate with respect to $\mathbf{Q}$: \...
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1answer
80 views

Matrices Commuting with a Kronecker Sum

Throughout, let $A$ and $B$ be complex $m \times m$ and $n \times n$ matrices respectively. By $A \otimes B$, we mean the matrix formed from the Kronecker product of $A$ and $B$, and by $A \oplus B$, ...
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Derivative of matrix using Kronecker Product

Suppose $ A(p)$ and $B(p) $ are functions which map $ \mathbb{R}^{n\times m} $ to $ \mathbb{R}^{n\times m} $ and $$F(p)=S(I_{q}\otimes A(p))(I_{q}\otimes B(p))M$$ where $ S,M $ are constant matrices ...
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Matrix representation of Mixed derivatives

Imagine we have the problem \begin{cases} -\frac{d^2u}{dx^2} = f(x), x \in [0,L] \\ u(0) = 0 \\ u(L) = 0 \end{cases} We know that we can approximate the second derivative using this ...
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Are the same expression kronecker Product and vec operator representation when differentiating a matrix by a matrix?

When differentiating a matrix by a matrix it is true that dA(X)/dX is the same thing as dvec(A(X))/dvec(X) acorrording to this paper. So, I compared kronecker product representation with vec ...
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1answer
66 views

How to find A and B in a kronecker product of A and B

I am studying a Kronecker product to solve a task in signal processing. it is a full correlation matrix of the channel. $$R=\begin{bmatrix} 0,8 &0,8&0,8&0&0&0\\ 0,8 &0,8&...
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How to determine the rank of a Khatri-Rao product of two matrices based on their each rank

As is known to all, the Khatri-Rao product is defined as $\mathbf{C}=\mathbf{A}\odot \mathbf{B}=\left[\begin{matrix}\mathbf{a}_1\otimes\mathbf{b}_1&\mathbf{a}_2\otimes\mathbf{b}_2&\cdots \...
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A question about Kronecker Product [closed]

I can' t show the following equality. Could you give some hints? $$ A^T\big( \psi(t) \otimes I \big) Q \big(\psi^T(t) \otimes I\big) A= A^T\big( \psi(t) \psi^T(t) \otimes Q\big) A$$ where $\otimes$ ...
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A question about spectral norm and kronecker product

Assume $A$ is a $n\times n$ matrix and $B=\left[\begin{matrix}B_{11}&B_{12}\\B_{21}&B_{22}\\\end{matrix}\right]$, then do we have $$\|A\otimes B\|=\|\left[\begin{matrix}A\otimes B_{11}&A\...
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trace inequality with symmetric Kronecker product

Let $A, B$ be two positive semi-definite $n \times n$ matrices and let $L$ be an $n \times n$ matrix that satisfies $\rho(L) < 1$, where $\rho(\cdot)$ denotes spectral radius. Let $A \otimes B$ ...
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41 views

Efficient Kronecker Product Formulation

Let $A$ and $B$ be two $p \times p$ matrices, where $p$ can be large. I am interested in finding $C$, where $$vec(C) = (I_{p^2} - A \otimes A)^{-1}vec(B)\,. $$ Here $\otimes$ denotes the Kronecker ...
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Is the Kronecker sum an affine transformation?

Is the Kronecker sum $ A \oplus B = A \otimes I_b + I_a \otimes B$ an affine function? Also, if so, would the following function also be an affine transformation $f(A,B) := A \otimes M_1 + M_2 \...
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Simplify bra-ket notation with kronecker product and kronecker sum

I am taking a quantum informatics and communication course, this is the first time I have faced with Dirac's Bra-ket notation. I have the following equation(Swap gate with 3 cnot): First equation $|...
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Kronecker product SVD Error bound

The Kronecker Product SVD (KPSVD) is defined here. Given a target rank $r$, what is the error bound in terms of singular values $\sigma_i$ for $\|A - A_r\|_F$, where $A_r = \sum_{i=1}^r \sigma_iU_i \...
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1answer
134 views

Derivative of a matrix using Kronecker product

I have this Equation (1) and I want derivative in respect to $\underline s $: $$diag(\underline s)\times C \times diag(\underline s)$$ that $\underline s $ is a $n \times 1$ vector and $C$ is an $n \...
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Linearly independence on Kronecker products

Let $R$ be a commutative ring with units. Suppose that $\{A_i \}$ and $\{B_j\}$ are two linearly independent families of $n \times n$ matrices over $R$. Is it true that the set $\{A_i \otimes B_j \}$ ...
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Eigenvalues of a (non-traditional) Kronecker sum

I'm trying to find some properties about the eigenvalues of the following operation. Let $A \text{ and }B\in\mathbb{R}^{n\times n}$ and consider $$ M = (\mathbb{I}_{\nu} \otimes A) + (B \otimes \...
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How do we calculate the trace over the matrix logarithm $\log((\sigma_2 \otimes I_{n/2})^T\cdot\Omega_{S^n})$?

How do we explicitly compute the curvature form $\Omega$ of the Levi-Civita connection $\nabla^{L.C.}$ for the $n$-sphere $S^n$? Thus, how do we calculate the trace over the matrix logarithm $\log(...
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Space-Efficient Calculations using Symmetric Kronecker Product: $(A\otimes_sB)^{-1}x$

Introduction: The Kronecker product of two matrices is defined as: $$\mathbf{A} \otimes \mathbf{B} = \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots &...
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1answer
83 views

Is there a way to compute $(A\otimes B)x$ quickly without forming the Kronecker product?

Is there a way to compute $(A\otimes B)x$ quickly without forming the Kronecker product? Often, I'd like to compute the matrix-vector product of a Kronecker product, but I'm not sure of a good way to ...
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How to find the inverse of $(A\otimes A)(B\oplus B)^{-1}(A\otimes A)+I$ without forming the Kronecker product?

Is there a way to invert $(A\otimes A)(B\oplus B)^{-1}(A\otimes A)+I$ without forming the Kronecker product? Here, both $A\succ 0$ and $B\succ0$. Generally speaking, I would say that finding the ...
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1answer
33 views

If $0 \leq A \leq B$, can I conclude that $(A \otimes I) \leq (B \otimes I)$?

Let $A \in \mathbb{R}^{n \times n}, B\in \mathbb{R}^{n \times n}$ are two positive semi-definite matrices and $A \leq B$ means $B - A$ is positive semi-definite. Also let $I \in \mathbb{R}^{m \times m}...
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1answer
56 views

vectorization of matrices

$vec(\boldsymbol{\beta}\boldsymbol{\sigma}_{n-1}^2\boldsymbol{e}_j'\boldsymbol{\sigma}_{n-1}^2\boldsymbol{\alpha}_1^\prime)=((\boldsymbol{\alpha}_1\boldsymbol{e}_j')\otimes\boldsymbol{\beta})vec(\...
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1answer
105 views

How to derive the following Kronecker product differential equation?

My question is from the following paper: https://www.nature.com/articles/ncomms5079 (equation (2) and (4)) I have the following: $$\delta\dot{\mathbf{x}}(t) = \bigg[\sum_{m=1}^M E^{(m)}\otimes D\...