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 $$\mathbf{A}\otimes\mathbf{B}=\mathbf{C}_{(KM\times LN)}=\begin{bmatrix}a_{11}\mathbf{B} &\dots & a_{1L}\mathbf{B}\\\vdots &\ddots&\vdots\\a_{K1}\mathbf{B} &\dots & a_{KL}\mathbf{B}\end{bmatrix}$$
412
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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 ...
3
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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 ...
1
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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:
$$
\mathbf{J}=\left[ \begin{array}{c}
\mathbf{I}_L\otimes \left[ \begin{matrix}
1& 0& \cdots& 0\\
\end{...
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vote
1
answer
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Matrix equation solving involving vectorization and Kronecker product
I am looking an analytical solution for a least-squares solution of the following equation for $X$:
$(A \ (I_{J} \otimes X)) \ \textrm{vec}(X) = b, $
where $A \in \mathbb{R}^{I\times J^2}$, $I_J$ is ...
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How can I generalize a vector-matrix-vector product into a vector?
I've been trying to generalize a vector-matrix-vector product that represents the $i$th element of a vector $v$, but I can't figure out how to put it into a concise form.
Let $v \in \mathbb{R}^{n}$ be ...
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Does the Kronecker product for matrices satisfies the universal property of the tensor product (of modules)?
The Kronecker Product is defined here: https://en.m.wikipedia.org/wiki/Kronecker_product
It is sometimes also called 'tensor product'.
Hence, I would like to know whether it satisfies the universal ...
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votes
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Equivalence of two constructions for $RM(r, m)$
I am currently studying about binary Reed-Muller codes, where we have first defined by the usual Boolean function definition:
Def The binary Reed-Muller code $RM(r, m)$ of order $r$ and length $2^m$ ...
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Does this permutation matrix have a name?
Below are examples of the matrix I have in mind:
$$
P_{2} = \begin{bmatrix}
1 & & & \\
& & 1 & \\
& 1 & & \\
& & & 1
\end{...
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0
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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 ...
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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 $n$....
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A generalization of the concept of eigenvalues and eigenvectors with kronecker products
Let $A$ be a $n \times n$ matrix. We know that $\lambda$ is an eigenvalue of $A$ and $x$ is its associated eigenvector if together they satisfy $Ax = \lambda x$. We can also write this equation as $Ax ...
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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
$$\mathop{\rm{min}}_ {\| \boldsymbol{v}_i \| = 1
\atop
i= 1,2,3} \| \boldsymbol{w} - \boldsymbol{v}_3 \otimes \boldsymbol{v}_2 \...
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Given a binary matrix M and a set of binary matrices S, how to decompose matrix M as kronecter product and/or multiplcation of matrices from set S?
Assume you are given a binary matrix $M$ (matrix elements are either 0 or 1) of higher dimension and a set of binary matrices $S = \{A_1, A_2, \cdots A_n\}$ of lower dimensions.
Now, I want to ...
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$\operatorname{vec}(AB) = \operatorname{vec}(A) \otimes \operatorname{vec}(B)$
I am looking for a more elegant way to confirm the following intuition:
Assume that $A$ and $B$ are two square $p\times p$ matrices. It seems there should always be some matrix $C$ such that $\...
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Hessian matrix of $\Lambda \mapsto y' (I + X\Lambda X')^{-1}y$
I have $$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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1
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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:
$$
A=
\begin{bmatrix}
\mathbf{x}^{\top} & \mathbf{0}^{\top} & \mathbf{0}^{\top}\\
\mathbf{0}^{\top} & \...
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What are the geometric multiplicities of the eigenvalues of a Kronecker (or tensor) product?
What are the geometric multiplicities of the eigenvalues of a tensor (i.e. Kronecker) product of linear transformations? We can assume that we are working in finite-dimensional vector spaces over an ...
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Solve matrix equations involving vectorization and Kronecker product
I want to find solutions for matrices $A\in \mathbb{R}^{m\times n}$ and $B\in \mathbb{R}^{n\times m}$ in the following equations:
$$
\left\{
\begin{matrix}
A^TM_1 = A^T(AB\odot M_2)
\\
...
- 13
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votes
0
answers
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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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Solving a matrix quadratic equation with diagonal restrictions - does this problem have an analytical solution?
The following matrices are all $n\times n$ and real.
$B$ and $D$ are unknown. Moreover, $D$ is diagonal, and $diag(D)=diag(B)$.
$\Sigma_{11}$, $\Sigma_{21}$ and $\Sigma_{22}$ are all known. $\Sigma_{...
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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 ...
- 6,974
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1
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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 ...
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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, ...
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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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Find eigenfunctions with kronecker product and sum
I need to find the eigenvalues of
$C+D$, where C is an nxn circulant matrix, and D is an nxn diagonal matrix. I know the eigenvalues $\lambda_i$ of $C$ and the eigenvalues $\mu_i$ of D. However, these ...
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Kronecker sum as a Kronecker product
I seek the following relationship (if there is one so):
𝐶⊗𝐷=(𝐴⊗I)+(I⊗D)
I would like to obtain A=𝑓(C,D)
We can assume D, circulant matrices, square and symmetric for simplicity (then all term of ...
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Scale blocks of rows of a matrix by different vector
Define vectors $\mathbf{a}_{i},\mathbf{b}_i\in\mathbb{R}^k,i=1,\ldots,n$. Define the $2 n\times k$ matrix $$\mathbf{C}=[\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{n},\mathbf{b}_{1},\ldots,\...
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Jacobian, vectorization and the Kroenecker product
Suppose
$f(U) = U^{T} A U$ with its derivative $d_{f}$ [U] $(H) = H^{T} A U + U^{T} A H$, then its Jacobian is given by
$J_f(vec U) = ((AU)^{T} \oplus I) \Pi + I \oplus U^{T} A$, (*)
where $\Pi = \Pi^{...
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On kronecker product representation of $AA^t$
Proposition: Prove that orthogonal matrices $\mathrm O_n(\mathbb{R})$ is a regular submanifold of $\operatorname{GL}_n(\mathbb{R})$.
Proof: Denote $f:\operatorname{GL}_n(\mathbb{R})\to \operatorname{...
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Can $[A\otimes I_m,B]=0$ for all $A$ imply $B=I_n\otimes C$?
Suppose that $A$ is an $n\times n$ matrix and $I_m$ is an $m-$dimensional identity matrix. If $[A\otimes I_m,B]=0$ for all $A$ where $\otimes$ is Kronecker product, does it imply $B$ must be of form $...
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If $\Phi$ is completely positive, then $\Phi\otimes\mathcal{I}$ is positve, is $\mathcal{I}\otimes\Phi$ also positive?
Suppose $\Phi$ is a linear map from $\mathbb{R}^{n\times n}$ to $\mathbb{R}^{n\times n}$, and satisfies that if $A\in \mathbb{R}^{n\times n}$ is positive semidefinite, then $\Phi(A)$ is positive ...
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Partial derivative of complex matrix products
I have a constraint optimization problem formulated in a diagonal matrix form:
$
P_3:~ min_{x} \quad \|A X(t) - Y(t)\|^2 \\
\text{subject to} \quad X^*(t) \cdot X(t) = \mathbb{I}
$
I need to ...
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Solve $A \otimes B + I = X_1 \otimes X_2$ [duplicate]
Is there a way to solve the following equation (find $\mathbf{X_1}$ and $\mathbf{X_2}$):
$$\mathbf{A} \otimes \mathbf{B} + \mathbf{I} = \mathbf{X_1} \otimes \mathbf{X_2},$$
where $\mathbf{A}, \mathbf{...
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Kronecker product, matrices
Let $\mathbb{I}_{3}$ be the $3\times 3$ identity matrix, i. e.
$\begin{pmatrix}
1 &0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}$
and $e_1:=\begin{pmatrix}
1 \\
0
\end{...
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Simplify ${\rm trace}((A_1⊗A_2+B_1⊗B_2)^{-1})$
This equation is take a long time in simulation due to Kronecker product ($⊗$).
Simplify $${\rm tr}((A_1⊗A_2+B_1⊗B_2)^{-1})$$
where tr is trace operator (sum of diagonal elements).
Also, $A_1,A_2,B_1,...
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Partition of Tensors
Let us say we have a Tensor
$$
\mathcal{X} \in \mathbb{R}^{J_1 \times J_2\dots \times J_N}
$$
Now we will partition this Tensor into 2 disjoint sets
$$
\mathcal{R} = \{r_1, \dots , r_L\}\\
\mathcal{C} ...
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votes
1
answer
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Determinant of Kronecker product $D = B⊗A$ is equeal $D=A^kB^n$
Here determinants:
$$
A = \begin{vmatrix}
a_{11} & ... & a_{1n}\\
... & ... & ...\\
a_{n1} & ... & a_{nn}\\
\end{vmatrix}
B = \begin{vmatrix}
b_{11} & ... & b_{1k}\\
......
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0
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Matrices determined by certain marginals with respect to the partial trace.
Suppose $A_i$ is a matrix algebra for $i=1,...,n$ (i.e., $A_i$ consists of $n_i\times n_i$ matrices with complex entries), and suppose $\tau$ and $\sigma$ are elements of $A_1\otimes \cdots \otimes ...
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Prove $tr_B(vec(A)vec(B)^T)=AB^T$
Let $A,B\in \mathbb{R}^{n\times n}$. And let $vec$ be a map : $\mathbb{R}^{n\times n}\to \mathbb{R}^{n^2\times1}$: $vec(e_a e_b^T)=e_a\otimes e_b$, here $e_a$ denotes the vector with with an entry $...
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vote
2
answers
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Kronecker product and vectorization of tensor modal product
Given a tensor $\mathcal{S}\in\mathbb{R}^{n_1\times\ldots\times n_d}$ and a matrix $M\in \mathbb{R}^{m_k\times n_k}$, let $\mathcal{A}$ be the modal product of $\mathcal{S}$ and $M$ along the $k$-th ...
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vote
1
answer
69
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Kronecker product between a matrix and a direct sum
Is there any way to simplify this matrix expression, using something like a distributive property?
$A \otimes \left( M_1 \oplus M_2 \right)$
where $\otimes$ is the Kronecker product and $\oplus$ is ...
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1
answer
54
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Linear Independence of a set of $n!$ vectors
Let $\{v_1,..,v_n\}$ be a linearly independent set of unit vectors in a finite dimensional real vector space. I want to show the set $\{v_{\sigma(1)}\otimes..\otimes v_{\sigma(n)}\}_{\sigma \in S_n}$ ...
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votes
1
answer
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Prove Corr$(\rho^{\otimes n},\mathcal{A},\mathcal{B})$= Corr$(\rho,\mathcal{A},\mathcal{B})^{\otimes n}$
Let $\mathcal{A}$ and $\mathcal{B}$ be two Hermitian basis of $m\times m$ matrices with complex entries. For example, when $m=2$, $\mathcal{A}$ can be $\{I,X,Y,Z\}$ where $X,Y,Z$ are Pauli Matrices. ...
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votes
1
answer
20
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Can the Kronecker product be applied when the distribution doesn’t result in scalar multiplication?
I just learned about the Kronecker product as a product of arrays. However, it seems that in all of the examples, when the Kronecker product is applied, the right side is multiplied with the ...
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votes
1
answer
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Does the Kronecker product preserve common irreducibility
A set of $d\times d$ real or complex matrices is called “commonly irreducible” if those matrices do not jointly preserve a linear subspace with dimension strictly between $0$ and $d$.
I wanted to know ...
