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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 tensor product preserving operations

Suppose I have two tensors, of order $m$ $$ A = \sum_{i=1}^p u_i \otimes \cdots \otimes u_i, \qquad B = \sum_{i=1}^p v_i\otimes \cdots \otimes v_i. $$ What are the class of transformations $T(u_i) = ...
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Can Kronecker Product Approximation (Nearest Kronecker Product) preserve invertibility?

Just as the title, I'm wondering about in Kronecker product approximation(Nearest Kronecker Product), if the original matrix is invertible, whether the two small matrices obtained after this ...
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Parial Derivative of a kronecker product

Let's say we have a function $f(x, A, B) = (A \otimes B)x$. We can assume A and B are matrices and x is some input matrix (it can be a vector). And I need to ...
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Linear matrix equation with stacked matrices and repeated unknowns

I have the following seemingly simple linear matrix equation, where I want to solve for the unknown matrix $K \in \mathbb{R}^{p \times m}$: $$ \begin{bmatrix} K A_0 \\ K A_1 \\ \vdots \\ K A_n \end{...
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Applying the Mixed-Product Property of the Kronecker Product to DFT and Identity Matrices of difference sizes

Suppose we have four matrices, $\mathbf{F}_{N_2}, \mathbf{F}_{N_1}, \mathbf{I}_{M_1}, \mathbf{I}_{M_2}$, where $\mathbf{F}_{N_1}$ and $\mathbf{F}_{N_2}$ are a DFT matrix of size $N_1$ and $N_2$, ...
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Gradient involving kronecker product of a vector and a matrix multiplied by another vector

Let $f(W):=AWx$ where $A$ and $W$ are matrices, $x$ is a vector. The gradient with respect to $W$ is given as: $$\nabla_Wf(W)=x^\top\otimes A$$ Now let $g(W):=y^\top AWx$, where $y$ is another vector. ...
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Is there an extension to this property of Kronecker products?

Fact: $$AXB = C \implies (B^T \otimes A)\text{vec}(X) = \text{vec}(C).$$ Now, let $A_1, A_2$ be symmetric, invertible matrices. This leads to a very useful identity (numerically), $$(A_1^{-1} \otimes ...
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Gradient of a complex-valued function with complex-valued variables

I have to minimize a cost function: $J = \frac{1}{2} e^* e$, where $e \in C$ is the error between the output of my ML model $y \in C$ and the desired value $m \in C$. Therefore, e is a complex number. ...
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Equivalent of logarithms for products of matrices

Let $a, b \in \mathbb{R}$. We know that $\log(ab) = \log(a) + \log(b)$. My question is very straightforward: Is there an equivalent function for one of the two most used matrix products? I.e., does ...
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Derivative (Jacobian) of a matrix equation

I have this equation: $y = e^{t(A + W)} x_0 $ where A is a diagonal matrix and W is a symmetric matrix. I need to find $\frac{\partial y}{\partial W}$. If A and W commute then I could use the fact ...
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Relation between pairwise outer products and Kronecker tensor product

Let $A,B$ be real $n\times n$ matrices (if we want complex entries, replace transpose with hermitian conjugate). Write them as an array of columns so that $A=\begin{pmatrix} A_1~|&\cdots& |~...
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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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Kronecker product and finite difference discretization for poisson equation

In this notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u =...
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Computing partial trace of a given kronecker product matrix with respect to the first component

Suppose I have two matrices, given as: $A= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $B= \begin{pmatrix} e & f \\ g & h \end{pmatrix}$ Then,the Kronecker product of $A$ and $B$ ...
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Inner product involving three PSD matrices

Let $\mathbb{S}^n_+$ denote the space of $n \times n$ symmetric positive semidefinite matrices. Let $A \in \mathbb{S}^n_+$ and $B \in \mathbb{S}^n_+$. Then $\langle A, B\rangle_F = \text{Tr}(A^\top B) ...
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Rewrite matrix operation involving diagonal matrix and Kronecker product

$D$ is a diagonal matrix in $\mathbb{R}^{(kn) \times (kn)}$, $W$ is in $\mathbb{R}^{n \times m}$. Does there exist a matrix $A \in \mathbb{R}^{n \times m}$ so that $D (W \otimes I_{k}) = A \otimes I_{...
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Kronecker Product Identity

Is there an Kronecker Product Identity that allows for factorisation? If $\textbf{A},\textbf{B}$ are $50\times50$ dim-matrices (B is diagonal), and $\textbf{C},\textbf{D}$ are $100\times100$ dim-...
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On the determinant of a Kronecker power

Let $$ Y := \left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \\ \end{array} \right)^{\otimes k} $$ where $\otimes$ denotes the Kronecker product. We have $$ \det(Y) = \left(\sqrt{2^k}\right)^{2^k} = 2^...
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Jacobian of a trajectory given by a matrix exponential

I need to get the jacobian of the function $x(t) = e^{At} x_0 $. so, I was thought about applying the vectorization and Kronecker product: $ d \, vec \, x = (x_0^T \otimes I) \, d \, vec (e^At)$. But ...
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Kernel of kronecker product of matrices

Consider a matrix $E \in \mathbb{R}^{m \times n}$ with $m\geq n$ and a nullspace $\text{ker}(E) = \{ \alpha 1_n , \, \alpha \in \mathbb{R} \}$, where $1_n$ is a column vector of ones of appropriate ...
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Prove Kronecker sum of two diagonalizable matrices is diagonalizable

Given two matrices $A$ (is $m \times m$) and $B$ (is $n \times n$) that are both diagonalizable, prove its Kronecker sum $$ A \oplus B := A \otimes I_n + I_m \otimes B $$ is diagonalizable, and give ...
Amulya Mohan's user avatar
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Is there a matrix that shift data along side the main diagonal?

I am an electrical engineer and I am trying to write my optimization problem in a more compact form. During my time solving this optimization problem, I reallize that I need to keep shifting and ...
Tuong Nguyen Minh's user avatar
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Weighted Nearest Kronecker Product

For a given $m n$-length vector $a$, the problem of finding an $m$-length vector $x$ and an $n$-length vector $y$ that minimize $$ \lVert a - x \otimes y \rVert^2 $$ is known as the Nearest Kronecker ...
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Can this matrix about the Kronecker product continue to be simplified?

There is a $ML \times ML$ matrix expression based on the Kronecker product: $$ \mathbf{J}=\left[ \begin{array}{c} \mathbf{I}_L\otimes \left[ \begin{matrix} 1& 0& \cdots& 0\\ \end{...
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Matrix equation solving involving vectorization and Kronecker product

I am looking an analytical solution for a least-squares solution of the following equation for $X$: $(A \ (I_{J} \otimes X)) \ \textrm{vec}(X) = b, $ where $A \in \mathbb{R}^{I\times J^2}$, $I_J$ is ...
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How can I generalize a vector-matrix-vector product into a vector?

I've been trying to generalize a vector-matrix-vector product that represents the $i$th element of a vector $v$, but I can't figure out how to put it into a concise form. Let $v \in \mathbb{R}^{n}$ be ...
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Does the Kronecker product for matrices satisfies the universal property of the tensor product (of modules)?

The Kronecker Product is defined here: https://en.m.wikipedia.org/wiki/Kronecker_product It is sometimes also called 'tensor product'. Hence, I would like to know whether it satisfies the universal ...
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Equivalence of two constructions for $RM(r, m)$

I am currently studying about binary Reed-Muller codes, where we have first defined by the usual Boolean function definition: Def The binary Reed-Muller code $RM(r, m)$ of order $r$ and length $2^m$ ...
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Does this permutation matrix have a name?

Below are examples of the matrix I have in mind: $$ P_{2} = \begin{bmatrix} 1 & & & \\ & & 1 & \\ & 1 & & \\ & & & 1 \end{...
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Does $(A \oplus A) X = b$ have a simpler solution via some identities?

Consider square matrix $A \in R^{n, n}$ and suppose we want to solve (like actually numerically solve) $$ (A \oplus A) X = b $$ where $\oplus$ is the Kronecker sum $A \oplus B = A \otimes I_B + I_A \...
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Hadamard product as Projection of Tensor/Kronecker Product?

The tensor product of matrices $A, B$ is the block matrix $$A\otimes B = \begin{pmatrix}a_{11}B & a_{12}B&\dots\\ a_{21}B&a_{22}B&\dots\\ \vdots & \vdots&\ddots \end{pmatrix}.$$...
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Is there literature on solving the linear system $\bf Ax=b$ without explicitly computing $\bf A$?

I am trying to solve a system of equations which can be represented in the well-known form of $\bf Ax = b$, where $\bf A$ is a large sparse matrix consisting of known values, $\bf x$ is a vector of ...
Robby Ram's user avatar
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Upper bound of biggest singular value of Kronecker Sum via singular values and traces

I am investigating the biggest singular value of Kronecker Sum. I'd like to understand if my argument is correct. Let $A$, $B$ be square matrices of size $n$. Let $I$ be an identity matrix of size $n$....
Piotr Lewandowski's user avatar
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Decomposes a unit vector into the kronecker product of three unit vectors

I want to get the optimal solution to the following equation $$\mathop{\rm{min}}_ {\| \boldsymbol{v}_i \| = 1 \atop i= 1,2,3} \| \boldsymbol{w} - \boldsymbol{v}_3 \otimes \boldsymbol{v}_2 \...
Jasper Cha's user avatar
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Given a binary matrix M and a set of binary matrices S, how to decompose matrix M as kronecter product and/or multiplcation of matrices from set S?

Assume you are given a binary matrix $M$ (matrix elements are either 0 or 1) of higher dimension and a set of binary matrices $S = \{A_1, A_2, \cdots A_n\}$ of lower dimensions. Now, I want to ...
FDGod's user avatar
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$\operatorname{vec}(AB) = \operatorname{vec}(A) \otimes \operatorname{vec}(B)$

I am looking for a more elegant way to confirm the following intuition: Assume that $A$ and $B$ are two square $p\times p$ matrices. It seems there should always be some matrix $C$ such that $\...
shadow1234's user avatar
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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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Localization of eigenvalues for block-tridiagonal Hermitian Toeplitz matrix made of gamma blocks

I am studying the spectrum of a particular kind of block-tridiagonal Hermitian Toeplitz matrix made of three bands $\{B,A,C\}$ $$ T_n = \begin{pmatrix} A & C & 0 & \dots & 0\\ ...
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Writing a sparse matrix as a Kronecker product

Let $\mathbf{x}=[x_1, x_2, x_3, x_4]^{\top}$. I was given the following matrix: $$ A= \begin{bmatrix} \mathbf{x}^{\top} & \mathbf{0}^{\top} & \mathbf{0}^{\top}\\ \mathbf{0}^{\top} & \...
Saeed's user avatar
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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 ...
Milten's user avatar
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Solve matrix equations involving vectorization and Kronecker product

I want to find solutions for matrices $A\in \mathbb{R}^{m\times n}$ and $B\in \mathbb{R}^{n\times m}$ in the following equations: $$ \left\{ \begin{matrix} A^TM_1 = A^T(AB\odot M_2) \\ ...
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Matrix valued system of equations with right and left product

Let $J$ be a real valued $2\times 2$ matrix with determinant equal to 1. The eigenvalues of $J$ are complex in general. Let $Q$ be a $2\times 2$ matrix satisfying $$J^T Q J = Q$$ Question: Does there ...
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Implementing Finite Differences solver for 2D Poisson Equation

I'm trying to solve the 2D Poisson equation: $$ \begin{cases} -\Delta u = f & \text{in} \hspace{0.2cm} \Omega=(0,1)^{2} \\ g = u & \text{on} \hspace{0.2cm}\partial \Omega \end{cases} $$ Using ...
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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 ...
Milten's user avatar
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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 ...
Vida Beach's user avatar
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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, ...
Milten's user avatar
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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{...
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