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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Matrix operation to exponentiate each element in a vector
I am using the following matrix algebra to obtain a vector, however, I eventually need all the resulting elements to be exponentiated.
\begin{equation}
\begin{split}
\boldsymbol{\beta}^{\textsf{T}}\...
user963084
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3
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Eigenspace of Kronecker power of a 2*2 rotation matrix
Let's consider a $2\times 2$ rotation matrix $R_\theta \in SO(2,\mathbb{R})$, and the following matrix obtained by repeatedly applying $n-1$ times the Kronecker product of $R_\theta$ with itself: $$Q_\...
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Find the relationship between $p$ and the number of solutions of this system, using the Kronecker - Capelli Theorem:
The system of equations:
$\begin{cases}-px+5y+3z=3\\2x-4y-z=p\\x+3py+pz=p\\\end{cases}$
This is how I learned to solve, using the Kronecker - Capelli Theorem:
Step1: Find the ranks of the coefficient ...
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how to calculate the derivative of a kronecker product $a^T(x\otimes c)$
fo vectors $x,a$ and $c$ how do we calculate the derivative of
$$a^T(x\otimes c)$$
with respect to $x$ where $\otimes$ denotes the kronecker product. Here we basically build one large column vector $(...
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Derivative of a triple product matrix
I am trying to find the solution for the following derivative
$$\frac{\partial \boldsymbol E\boldsymbol J\boldsymbol E^{T}}{\partial \boldsymbol E}$$
where $\boldsymbol E$ and $\boldsymbol J$ are both ...
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Deriving a simpler expression for $\left({I}-T\otimes T\right)^{-1}\left(T\otimes T\right)$ using Kronecker product properties
Let $T\in\mathbb{R}^{n\times n}$.
Is there a simpler expression for $$\left(I_{n^{2}}-T\otimes T\right)^{-1}\left(T\otimes T\right)~?$$
When $T$ is symmetric PSD, we could do use the unitary ...
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Is there a closed form for this geometric-like series of two matrices?
Obviously, if for a square matrix $A$ we have $|A|<1$, then $\sum_{i=0}^\infty A^i = (I-A)^{-1}$ which is defined. Obviously also holds for a product of two (same-sized, square) matrices $AB$ with ...
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How to notate off-diagonal blocks for a tridiagonal block matrix
I have a block diagonal matrix that also has partitioned identity matrices in the off-diagonal blocks, like a tridiagonal block matrix. It looks something like
$$
\begin{bmatrix}
\textbf{A} & I &...
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How is $\boldsymbol{v} \stackrel{q}{\otimes} \boldsymbol{v}$, the tensorial product of a vector by itself repeated $q$ times, is defined?
The tensorial product of a vector $\boldsymbol{v}$ by itself is given as $\boldsymbol{v} \otimes \boldsymbol{v} = \boldsymbol{v} \boldsymbol{v}^\mathrm{T}$ which is a tensor of dimension 2. While i ...
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How to understand notation $\otimes$ for multivariate Gaussian distribution?
I am confused about the notation $\otimes$ for multivariate Gaussian distribution. For $K=(K_{s,r})$ and $1\le s, r \le n$ a n by n covariance matrix, we write
$$(Z_1, Z_2, \dots, Z_n)\sim \mathcal{N}(...
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What is the formal name of this matrix product?
Given two $m\times n$ matrices, I want to calculate the sum of the inner product of every pair of rows (or columns) in the two matrices, therefore the result is a real number. This matrix operation ...
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Spectrum of a Kronecker sum of an operator with itself
Let $A$ be an arbitrary diagonalizable square matrix. Consider $A \oplus A = A \otimes I + I \otimes A$ that is a Kronecker sum of $A$ with itself ($I$ is of the same size as $A$ here). I want to ...
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Expected value of Kronecker product between two random matrices
Let $Y$ be a $n\times p$ random matrix with expected value $\mathbb{E}(Y)=M$ and variance $\mathbb{V}(vec(Y))=V \otimes U$, with $V$ and $U$ positive semidefinite $p\times p$ and $n\times n$ matrices, ...
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If $\sum_ix_i\otimes y_i=0$ and $x_i$ are independent, then prove that $y_i=0$ for all $i$. [duplicate]
Let $V$ and $W$ be two vector spaces such that $x_i\in V$ and $y_i\in W$. If $\sum_ix_i\otimes y_i=0$ and $\{x_i\}$ are linearly independent, then prove that $y_i=0$ for all $i$.
What I know is the ...
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Understanding the Unitary Operator $U|\psi\rangle\otimes|0\rangle=\sum_m M_m|\psi\rangle\otimes|m\rangle$
The operator U defined as $U|\psi\rangle\otimes|0\rangle=\sum_m M_m|\psi\rangle\otimes|m\rangle$ where $\sum_mM_m^\dagger M_m=I$, preserves the scalar product $(\langle \phi|\otimes\langle 0|U^\...
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Spectral decomposition of self kronecker products [duplicate]
Let $\vec{x} \in \mathbb{R}^n$ be a real column vector and $\mathbf{X} = \vec{x} \otimes \vec{x} \triangleq \vec{x}\vec{x}^T \in \mathbb{R}^{n \times n}$ the Kronecker product of $\vec{x}$ with itself....
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Can a vector be decomposed (non-uniquely) into tensor products of smaller vectors?
I seek an extension of the solution found in this question, which refers to the "Nearest Kronecker Product".
Given $A\in \mathbb R^{m\times n} $ with $m = m_1m_2m_3$ and $n = n_1n_2n_3$, ...
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Relationship between Kronecker square and vec square
Let $X$ be a $n$ by $n$ symmetric matrix. Let vec$(X)$ denote the operator that stacks the columns of $X$ into a row vector and let $\otimes$ denote the well-known Kronecker product.
The two $n^2$ by ...
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One more question about matrix calculus: this time about skew-symmetric matrices
Formula for the transition from tensors to vectors in matrix calculus
https://mathematica.stackexchange.com/questions/251079/derivative-matrix-by-vector-in-mathematica
Given:
$Q=R_z(\psi)R_y(\xi)R_x(\...
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Derivative of a scalar quantity involving inverse and Kronecker product
How can we compute the derivative of the following quantity with respect to $\bf{\Sigma}$?
$$
\phi = {\bf{x}}^\top({\bf \Sigma^{-1}\otimes I){\bf x}}
$$
Edit: I have tried to solve this problem by ...
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Formula for the transition from tensors to vectors in matrix calculus
Given quadratic norm:
$f=\boldsymbol{g}^T\boldsymbol{p}=\boldsymbol{g}^TQ\left[\begin{array}{@{}c@{}} 0 \\ 0 \\ 1 \end{array} \right]$
where $\boldsymbol{g}=\left[\begin{array}{@{}c@{}} 0 \\ 0 \\ -9....
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Extremely complex vector-matrix expression and its differentiation by vector
Given:
$Q=R_z(\psi)R_y(\xi)R_x(\phi)$ - rotation matrix
$\boldsymbol{\theta}=\left[\begin{array}{@{}c@{}} \phi \\ \xi \\ \psi
\end{array} \right]$ - vector of angles
$p=Q\left[\begin{array}{@{}...
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How to decompose a matrix as the sum of Kronecker products?
I encounter a problem revalent to Kronecker product (KP).
I want to decompose $A=\sum^r_{i=1}B_i\otimes C_i$, where $A\in \mathbb{R}^{8\times2}, B_i\in \mathbb{R}^{4\times2}, C_i\in \mathbb{R}^{2\...
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How to prove this inequality about Kronecker product?
I encounter the following problem when I study ridge regression.
Problem. Let $\{d_j\}_{j=1}^\infty$ be a sequence of positive integers. Let $\{\psi_{i,j}\}_{i,j=1}^\infty$ be a collection of vectors ...
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Inverse of the sum two Kronecker product
We know if we have two matrices $A$ and $B$ then:
$(A\otimes B )^{-1} = A^{-1} \otimes B^{-1}$
Now if we have four matrices $A,B,C,D$ . Is there an equivalent term for the following?
$((A\otimes B)+ (...
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Wikipedia's property 5 of Khatri-Rao products
I ran across the face-splitting product recently, but there is a frequently mentioned property that I cannot make sense of. The property is Property 5 at Wiki's Khatri-Rao page:
$$\mathbf{A} \otimes (\...
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Kronecker product vec trick but middle matrix is transposed
There are famous Kronecker product trick
$$
vec(AXB) = (B^T \otimes A)vec(X)
$$
where $vec(X)$ - stacks columns of $X$. It's super useful when we need to take derivatives of expression with respect to ...
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CP decomposition as a special case of Tucker decomposition
I am reading this article "Tensor Decompositions and Applications" by Kolda and Bader. On page 21, it says:
...CP [decomposition] can be viewed as a special case of Tucker [decomposition] ...
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PCA, Kronecker product
Let $\mathbf{A}\in\mathbb{C}^{m \times n}$ and $\mathbf{B}\in\mathbb{C}^{m \times n}$. Find the solution to the following problem:
$\min_{\mathbf{X} \in \mathbb{C}^{{m}^{2} \times r},~~ \mathbf{Y} \in ...
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Equality of matrix commutation within the trace operation
I am working on one specific problem, which mounts to prove the following equation. Say $a\in\mathbb{R}^n,b\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. In what situation does the following ...
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The Kronecker product of two $A_4$ irreducible representations and a decomposition into its symmetric and antisymmetric parts
From the character table of $A_4$, I understand the following Kronecker products hold:
\begin{align}
{\bf 3} \times {\bf 3} = {\bf 1} + {\bf 1}_1 + {\bf 1}_2 + {\bf 3} + {\bf 3}.
\end{align}
However, ...
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Derivative of F-norm with Kronecker product
For the function:
$f=\frac{\alpha}{2}\|A(I_{d}\otimes X)B\|_{F}^{2}+\frac{\beta}{2}\|X-Z\|_{F}^{2},$
where $X,Z\in\mathbb{R}^{m\times n}, A\in\mathbb{R}^{k\times dm},B\in\mathbb{R}^{dn\times\ell}$. ...
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Partial derivative involving Kronecker product
I have the following question:
For the function
$f=\|XY^\top-A(I\otimes X)Z^\top\|_{F}^{2}$
where $\otimes$ denotes the Kronecker product, and $\|\cdot\|_{F}$ denotes the Frobenius norm.
What is the ...
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Simplifying a Hessian
Let $b: \mathbb R^m \to \mathbb R$ be a differentiable function, $A \in \mathbb R^{d \times m}\;$ and $x \in \mathbb R^d$. Consider the function $f(A) = b(A^T x)$. Let us denote the argument of $b$ as ...
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Vector - Matrix Differentiation that includes the Kronecker product
I have that ${y}={A}\otimes{A}{x}$ where ${A}\in\mathbb{R}^{n\times n}$ and ${x}\in\mathbb{R}^{n^2}$. I want to find $\frac{d{y}}{d{A}}$ in matrix (or tensor) form. I have looked at other questions on ...
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How to find the Jacobian and Hessian of a function involving multiple Kronecker products?
I am having trouble finding the Jacobian and Hessian of this function involving the Kronecker product. I have a matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ and a vector $\mathbf{x}\in\mathbb{R}^{n^4}$...
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A useful transformation for Markov Chain using matrix operations.
I'm working with Markov chains in Python and I need to transform the transition matrix in order to facilitate some statistical estimation and I don't know what kind of matrix operations I could use.
...
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Prove that $\mathrm{Tr}(B^\mathsf{T}Y^{-1}B)$ is independent of $B$
Given diagonal $A\in\mathbb{R}^{n\times n}$ with all eigenvalues larger than $1$, and minimal polynomial $\alpha(\lambda)$.
Matrix is called cyclic if its minimal polynomial is equal to characteristic ...
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Vector spaces and Kronecker product
Can we express any vector $z \in \mathbb{R}^{nm}$ as the Kronecker product of elements of $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^m$?
I was working a little example where $n=m=2$ and we can write the ...
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Matrix identity involving Kronecker product
It is well-known that, given real matrices $\mathbf{A}_1$, $\mathbf{A}_2$ and $\mathbf{B}$ with dimensions $m_1 \times n_1$, $m_2 \times n_2$ and $n_1 \times n_2$, respectively, then
$$(\mathbf{A}_2 \...
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Show an objective function is convex with respect to multiple matrix variables
Before go through the details, you can just first consider a toy example $(m,n,N,d_1,d_2)=(2,3,4,1,1)$ which is the simplest case.
I'm solving an optimization problem numerically(probably with Newton'...
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Eigenvalues of Kronecker product of two non-square matrices
Let $\mathbf{A}\in\mathbb{R}^{m\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times m}$ be rectangular matrices. What can we say about the eigenvalues of their Kronecker product $\mathbf{A} \otimes \...
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Time complexity of Kronecker product $I_n \otimes Q$ where $Q$ is Toeplitz
Given a symmetric Toeplitz matrix $Q$ of size $m$, what is the time complexity of the Kronecker product $I_n \otimes Q$, where $I_n$ is identity matrix of size $n$? Is is $\mathcal{O}(m)$?
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How to show the hermitian adjoint of two operator's kronecker product is equal to their hermitian adjoints' kronecker product?
I mean, the question is simple, asking me to show $(A\otimes B)^* = A^*\otimes B^*$, where * is just the hermitian adjoints. However, it's hard for me to use the definition, aka put them into (x,y) ...
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Matrix exponential of a Kronecker product
I'm trying to find an expression for the matrix exponential of a Kronecker product of two matrices, $\hat{c}$ and $\hat{D}$. The matrix $\hat{c}$ is a small real and symmetric $2\times 2$ matrix:
$$
\...
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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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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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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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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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