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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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Flipping Summation of Kronecker Products

Question Suppose $\mathbf A$ is an $n\times n$ matrix, and that $\mathbf B_i$ is an $m\times m$ matrix, for all $i\in\{1,\dots, n\}$. Is it possible to find $n\times n$ matrices $\mathbf U$ and $\...
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How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
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computing the inversion of a matrix which is the sum of a Kronecker product and an identity matrix

I'd like to evaluate a single entry $s_{ik}$ of the $\mathbf{S}$ matrix, using Markov chain Monte Carlo approach. The posterior of $\mathbf{S}$ has a Gaussian likelihood with a covariance matrix $$\...
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Is there any specific criteria for a matrix $A(t)$ to have either time-dependent or independent eigenvectors?

I am investigating properties of a matrix $$A(t_1,t_2) \equiv U_1(t_1) \otimes U_2(t_2) - U_2(t_2) \otimes U_1(t_1)$$ where $U_1$ and $U_2$ are time-dependent unitary matrices. I'm finding that for ...
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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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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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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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How to find the inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ without forming the Kronecker product?

Is there a good way to compute the inverse of Inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ that doesn't require forming the full Kronecker product? Here $A$ is symmetric, positive ...
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Expectation of double quadratic form

I'm reading the 2012-version “The Matrix Cookbook”. On Page 43 Section 8.2.4 “Mean of Quartic Forms” there is a formula that really confuses me: $E[x^TAxx^TBx]=Tr[A\Sigma(B+B^T)\Sigma]+m^T(A+A^T)\...
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kronecker product of three matrices

Facts: For matrices $A_i\in \mathbb{R}^{n\times n}$ with $i=1, 2, 3$, we have the following equation: $$ A_1\otimes A_2 \otimes A_3 = (A_1\otimes I_{n^2})(I_{n}\otimes A_2 \otimes I_n)(I_{n^2}\otimes ...
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Matrices with the sign pattern of the Kronecker sum

The Kronecker sum is defined as $$ A \oplus B = A \otimes I_m + I_n \otimes B, $$ where $\otimes$ is the Kronecker product, and $A$ is $n\times n$ and $B$ is $m \times m$. It has lots of nice ...
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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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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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Bounding the determinant of principal sub-matrices of the Kroneker product

I have a matrix $A$ that is 2 dimensional and has negative determinant. I have a matrix $B$ that is 2 dimensional and has a positive determinant. Both have strictly positive elements. I want to show ...
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Inverse of the summation of Kronecker products with positive definite matrices

I would like to to obtain the inverse $\textbf{C}^{-1}$ of a matrix having the following form: $$\textbf{C} = \sum_{k=1}^K A_k \otimes B_k + I,$$ where $I$ denotes the identity matrix and where $\...
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Rank of kronecker product of partitioned matrix

Assume that there is a $NL \times K$ matrix $G=\begin{bmatrix} G_1 \\ G_2 \\ \vdots \\ G_N \end{bmatrix}$ such that $rank(G)=K$ and another $NM \times Q$ matrix $H=\begin{bmatrix} H_1 \\ H_2 \\ \vdots ...
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378 views

Vector Multiplication with Multiple Kronecker Products

My question concerns matrix-vector multiplications when your matrix has Kronecker structure, which can be done faster in that case. I know how to compute this for a matrix $A = A_1 \otimes A_2$, ...
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812 views

Eigenvalues of a Kronecker Product type matrix

We have matrix $C$ of the form: $C =\begin{bmatrix} B_{1,1} A_{1,1} & B_{1,2} A_{1,2} & \dots & B_{1,K} A_{1,K} \\ B_{2,1} A_{2,1} & B_{2,2} A_{2,2} & \dots & B_{2,K} A_{2,...
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How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?

The mode-$n$ product of a tensor $\mathcal{X}=[x_{i_1,\ldots,i_M}]\in \mathbb{R}^{I_1\times \cdots \times I_M}$ and a matrix $\mathbf{U}=[u_{i_m,j}]\in \mathbb{R}^{I_m\times J}$ is denoted by $\...
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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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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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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 to simplify notation for the Kronecker product of multiple matrices?

I would like to know if I can simplify notation for the Kronecker product of multiple matrices: $$A = A_1 \otimes A_2 \otimes \cdots \otimes A_n$$ For example, I could use a product sign such as: $$...
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Convergence of series of matrices with kronkecker product inside it

I've got trouble finding the convergence of the following series. Let's first assume that: $$ \lim_{n\rightarrow \infty} \theta_{0}^{n} = 0$$ The series I've got trouble finding the convergence of is: ...
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Writing this matrix expression in terms of vec operator

Let $X$ be a $p \times k $ matrix with $p > k.$ Is there a natural way to write either $$\operatorname{Tr}\left\{C^T X(X^T X)^{-1/2}\right\}$$ or $$\operatorname{Tr}\left\{B\left[X(X^TX)^{-1/2}\...
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How to decompose a vector into a sum of subvector with kronecker product?

Define a random $15 \times 15$ square matrix $\bf{A}$ which is full rank. If ${\bf{w}} = vec({\bf{A}})$, I want to know how to decompose $\bf{w}$ by subvector via kronecker product, i.e., ${\bf{w}} = ...
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Multivariate Least Squares using the Kronecker product.

I am working in a multivariate linear regression setting with the following model: $$Y_{n \times q} = X_{n \times p} B_{p\times q} + \mathcal{E}_{n \times q}.$$ Let $\mathcal{E}$ have variance-...
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153 views

Non-linear matrix equation solvable with linear algebra?

Consider the matrix equation $${\bf X}^k\bf {A = B}$$ Which we want to solve for $\bf X$ We can put A and B in a "block-vector": $v = [{\bf A}^T,{\bf 0},\cdots,{\bf 0},{\bf B}^T]^T$, assume there ...
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Symmetrized Kronecker Product

Define $\operatorname{ctri}_N:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}^{N\times N}$ by: $${\operatorname{ctri}_N{(a,b)}}_{i,j}=\begin{cases} a, && j>i \\ \frac{a+b}{2}, && j=...
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Help with a derivative involving Kronecker product

I have the following matrix form, $$\tilde{R}=[(I_T\otimes V)+(I_T\otimes\Sigma(\theta))K^{-1}(I_T\otimes\Sigma(\theta))']$$ where $\theta=[\theta_1,...\theta_N]$ is a vector, so each element of the ...
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199 views

Relate Direct Sum and Kronecker Product of Matrices

Is there a general relation between the direct sum of matrices $A \oplus B = \mathrm{diag}(A,B)$, yielding a block diagonal matrix, and the Kronecker product $(A \otimes B)_{ij} = a_{ij} B $? For ...
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Inverse Kronecker product or minimize a distance norm

Let $A$ be a real valued $4n \times 4n$ matrix, $B$ is a real valued $4\times4$ matrix, and $C$ is a non-negative $n\times n$ matrix. I have $A$ and $B$ and I am trying to get $C$ through this ...
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218 views

inverse of Tracy–Singh product

We know that $(A⊗B)^{−1}=A^{−1}⊗B^{−1}$ for kronecker product Is this true for Tracy–Singh product or Khatri–Rao product which is a kronecker product of partitioned matrices. See wiki kronecker ...
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173 views

Eigenvalues of a matrix formed by tensor multiplicataion

I am dealing with a matrix $H\in\mathbb{R}^{N \times N}$ that is the result of a vector $S\in\mathbb{R}^{1\times P}$ multiplied onto an order-3 tensor $T\in\mathbb{R}^{P\times N \times N}$. I am not ...
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137 views

Kronecker Product Reformulation

I have 2 questions concerning the reformulation of a Kronecker products where i kind of got stuck. Firstly assume the matrix $V\in \mathbb{C}^{N\times M}$ is defined as $$ V=\left[\begin{...
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112 views

Product of matrix-valued normal densities and Kronecker product

I am trying to find an expression for the mean, column-covariance and row-covariance matrices of the product of two matrix-valued Normal distributions. Here is what I've tried in a special case I ...
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334 views

Writing an expression in terms of vectorization operator $\mbox{vec} (X)$

I am new with vectorization and Kronecker products. I need to write the scalar value $$\mathbf{a}^{T}\mathbf{X}\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{a}$$ in terms of $\...
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Efficient evaluation of quadratic form for a Kronecker-decomposable covariance matrix?

Well, it's in the title. I'd like to evaluate $\mathbf{y}'K^{-1}\mathbf{y}$ in the case that $K$ is decomposable into a Kronecker product (e.g. $K = A\otimes B\otimes C$). (EDIT: $K$ is also symmetric ...
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412 views

Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
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67 views

Loop integral with Kronecker-delta notation

I worked through a problem and arrived at the final solution (which is correct), however, one part of it should equal zero mathematically. This is the part that should equal zero: $F_{2i}= (\frac{...
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86 views

Quadratic form of Kroenecker products of skew-symmetric matrices

I am trying to understand under which conditions on $P=P^\top>0$ , $C=C^\top$, the following quadratic form is zero: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
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105 views

an inequality about kronecker product with eigenvalues question

Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of $0=\lambda_{1}(L)<\lambda_{2}(L)\leq...\leq\...
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8 views

Taylor expansion using Kronecker tensor

I have the following function (just used as an example): $y_t=g(y_{t-1},\epsilon_t, \sigma)$ of which I have the following second-order Taylor expansion around a point such that $y=\bar{y}, \ \...
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45 views

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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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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18 views

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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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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27 views

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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70 views

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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23 views

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 &...