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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Find decomposition of a kronecker product

I know that $\underbrace{A}_{mn \times mn} = \underbrace{B}_{n \times n} \otimes \underbrace{C}_{m \times m}$. Both A and B are know symmetric and positive definite matrices (they are covariance ...
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Can we write down $\mathrm{vec}(Y^{-1})$ in terms of $\mathrm{vec}(Y)$?

Lets denote the Kronecker product by $\otimes$ and the vectorization of a matrix $Y$ by $\mathrm{vec}(Y)$. Given $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$, where $n\geq m$. What ...
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Equivalence of kronecker product of matrices and tensor product of vectors.

I know how kronecker product of matrices $A\otimes B$ where $A,B\in M_n(\mathbb{R})$ and tensor product of vectors $x\otimes y=xy^T$ are defined. I am perplexed about the equivalence of both the ...
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Give an example of a symmetric matrix $A$ such that $A\in S_4^+$ but $A\neq \sum_{k=1}^NX_k\otimes Y_k$ where both $X_k,Y_k\in S_2^+$.

I have trouble in finding an example of a symmetric matrix $A\in M_4(\mathbb{R})$ such that $A$ is positive semidefinite matrix but $A$ cannot be written as sum of tensor product of matrices $X_k,...
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Column-wise partitioned matrix multiply by Kronecker Sum matrix

I saw one operation (stated below) in a proof and I don't think I completely understand its innate operation logic although I can guess the answer. The proof idea is: Suppose $\exists$ a non-singular ...
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Null Space of fix point equation where coefficient matrix is a Kronecker Product

I am interested in finding the null-space of the fix point equation: $$ Pw = w \\ (P_{out} \otimes P_{in}^T)w = w $$ Where $P \in \mathcal{R}^{nm \times nm}$ is the Kronecker product of two ...
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Kronecker powers and k-ary words

Let $A$ be any $k\times k$ matrix. Also, let $\otimes$ denote the Kronecker product and define $A^{\otimes n},$ the $n$th Kronecker power of $A$, by $A^{\otimes 1}:=A$ and $A^{\otimes n}:=A\otimes A^{\...
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Trace of Kronecker product $tr((I_N\otimes \Sigma_2)(\Sigma_1\otimes I_N))$

Let $\Sigma_1, \Sigma_2\in\mathbb{R}^{d\times d}$ symmetric and positive definit and $N\in\mathbb{N}$, $N\neq d$. Can the following term be simplified? $$tr((I_N\otimes \Sigma_2)(\Sigma_1\otimes I_N))$...
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Derivative of complex quadratic form-functions with respect to a vector

I have a quadratic form like this: $Q_1=v_1^TC_1(x)v_1$ $Q_2=v_2^TC_2(y)v_2$ where $x,y,z$ - $3 \times 1$-vectors $J_1,J_2,C_1,C_2$ - $3 \times 3$-matrix $v_1=x-(J_1(x)y-z),v_2=x-(J_2(y)y-z)$ ...
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Frobenius norm inequality for product of matrices with kronecker product structure

Consider three integers $p,n,t$. Consider a matrix $M\in R^{pt\times pt}$ symmetric positive definite with eigenvalues at most one of size $(pt\times pt)$, a matrix $A\in R^{t\times t}$ and a matrix $...
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Coherence Upper Bound of Kronecker Product and Transposed Khatri-Rao Product

When I am doing some research projects, I am trying to lower the total coherence of a matrix which is defined as $$ \mu^t(\mathbf{Q})=\sum_{m=1}^G\sum_{n=1,n\neq m}^G\left(\mathbf{Q}(m)^\mathsf{H}\...
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Prove that $\delta_{jl}\delta_{im}=\delta_{jm}\delta_{il}$

Prove that $\delta_{jl}\delta_{im}=\delta_{jm}\delta_{il}$ In the video, he directly cancelled $$3\delta_{jl}\delta_{im}-3\delta_{jl}\delta_{im}$$ and similar terms. I was thinking if subscript ...
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Shorter (more concise) writing of a certain block matrix

Given a $n\times n$ matrix $U=[u_{ij}]$ and if we denote with ${\bf u}_k$ its columns ($n\times 1$ matrices), I wonder if there is a way to write the following $n^2\times n^2$ block matrix in an ...
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Kronecker Product of Normal Matrices

Theorem. If $A\in M_m$ and $B\in M_n$ are both normal, so is $A\otimes B$. The converse is true if $A\otimes B\ne 0$. Proof. Suppose that $A\in M_m$ and $B\in M_n$ are both normal. Then \begin{align*}...
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Kronecker product rule

Let $A\in\mathbb{R}^{p\times q}$, $B\in\mathbb{R}^{r\times s}$, and $C\in\mathbb{R}^{pr\times qs}$. Consider: \begin{equation} (A\otimes B)C^{\top}. \end{equation} I'm wondering if there are any rules ...
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Vectorization of the Extended Kalman Filter Gain Equations

I was trying to vectorize the extended Kalman Filter equations, I got the Kalman gain equation vectorized and got it working for a very niche case (where only 1 measurement is available). Originally: $...
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Vector-matrix differentiation and vectorisation

In recurrent neural network backpropagation (BPTT), we have the equations: \begin{align} e_t &= E^T x_t \\ a_t &= W_{hx}^T e_t+ W_{hh}^T h_{t-1}\\ h_t &= \text{tanh}(a_t) \\ s_t &= W_{...
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How do you take the derivative of a trace of matrix kroenecker products?

Let $M \in \mathbb{C}^{vr \times vc}$ and $P \in \mathbb{R}^{v \times v}$, where $P \succ 0 $ then, $F = Tr(M (P^{-1} \otimes I_{c}) M^H (P \otimes I_{r}))$, where, $I_{r} \in \mathbb{R}^{r \times r}$ ...
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Explanation of mixed product of tensors using basis vectors

I cannot understand the mixed product $$ (A\otimes B) (C\otimes D) = (AC) \otimes (BD) $$ I try to follow from the bases: $$ \begin{eqnarray} \left(A\otimes B \right) \left(C \otimes D \right) &=&...
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Kronecker product identity when multiplied by two vectors?

I am interested in least-squares optimization for problems with space-time separable prior state covariances and am trying to break down the quadratic cost function into respective space-time ...
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Pseudo inverse (left inverse) of $(I_n \otimes v^T) + (v^T \otimes I_n) $

Consider a column vector $v\in \mathbb{R}^n$. We are interested in finding the pseudo-inverse of the following matrix: \begin{align} A= (I_n \otimes v^T) + (v^T \otimes I_n) \end{align} where $I_n$ ...
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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}}\...
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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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A question about Kronecker product between a vector space $V$ and $\mathbb{R}^\alpha$

I have checked the definition of Kronecker product and I'm just trying to convince myself about the following: If I have a (finite-dimensional) vector space $V$, then $$V \otimes \mathbb{R}^{\alpha} = ...
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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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is the conjugate transpose of tensor (kronecker )product equal to tensor product it self?

just below equation 6 in https://sites.cs.ucsb.edu/~vandam/teaching/S05_CS290/mathnotes1.pdf, I read the conjugate transpose of tensor product equal to tensor product it self. I am looking for more ...
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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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How to express $\sqrt{A^TA}$ in terms of $A$

I have a Gramm matrix $G=A^TA$. Is there any simple way to express $\sqrt{G}$ in terms of $A$ using standard matrix operations, like product, transposition, inverse, element-wise function $f(A)$, ...
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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 combine two Dirichlet distributions via Kronecker product

I have two Dirichlet distributions, $(X_1, X_2=1-X_1) \sim \text{Dir}(\vec{\alpha_1})$ and $(Y_1, Y_2=1-Y_1) \sim\text{Dir}(\vec{\alpha_2})$, that model the probabilities associated with two Binomial ...
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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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