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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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170 views

Derivative involving the trace of a Kronecker product

I'm stuck trying to solve a derivative that looks like this: $$\frac{\partial}{\partial X} \mbox{Tr} \{ A(X^{-1} \otimes I_{n} )B \},$$ where A is a $N\times 2n$ matrix, B is a $2n \times N$ matrix, ...
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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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4answers
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Determinant of the Kronecker Product of Two Matrices

I'd like to know how can be shown that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ respectively and $\otimes$ represents the Kronecker product of ...
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1answer
332 views

Operator norm (induced $2$-norm) of a Kronecker tensor

Let $A \in \mathcal M(n \times n; \mathbb R)$ with $\rho(A) < 1$. Then we know $I \otimes I - A^T \otimes A^T$ is invertible where $\otimes$ denotes kronecker product. Let $\text{vec}$ denote the ...
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2answers
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Sum of two kronecker products as a kronecker product

I seek for the following relationship (if there is one so): $$C \otimes D = (A_1 \otimes B_1) + (A_2 \otimes B_2)$$ I would like to obtain $C = f(A_1,A_2)$ (in terms of $A$'s) and $D = g(B_1,B_2)$ (...
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0answers
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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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1answer
105 views

Matrix Differentiation of Kronecker Product

I have a question about differentiating an expression which has multiple kronecker products. I have the following objective function I would like to differentiate with respect to $\mathbf{Q}$: \...
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5answers
698 views

Invertibility of a Kronecker Product

Prove that $A\otimes B$ is invertible if and only if $B\otimes A$ is invertible. I don't have a clue where to start to be honest. I am not very familiar yet to the Kronecker Product so could you ...
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1answer
123 views

Finding Hessian of tr ((AB)' (AB))

I'm trying to find Hessian of $\text{tr}((AB)' (AB))$ where $A,B$ are matrices. There are nice expressions for $H_{AA}$ and $H_{BB}$ using standard approach from Magnus 1 , can anyone suggest how to ...
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1answer
321 views

Trace of Product of Kronecker Products

Start Wearing Purple answered this question for me. I am now asking a question about their answer as I can make the question general enough to be useful on it's own. For $1\times 2$ matrices $\alpha,\...
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2answers
126 views

Computing the derivative of $Axx^TB^T$ with respect to $x$

I want to compute the derivative of \begin{align} f(x) = Axx^\top B^\top \label{eqn} \end{align} with respect to $x$ where $A$ and $B$ are $n\times n$ matrices and $x$ is a (column) vector of size $...
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1answer
175 views

A question about matrix kernels and Kronecker products

Let us define $$ v:=v_A\otimes v_B\quad (*) $$ where $v_A$ is a fixed vector in $\mathbb{R}^{d_A}$, $v_B$ is any vector in $\mathbb{R}^{d_B}$ and $\otimes$ denotes the Kronecker product. To rule out ...
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0answers
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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1answer
131 views

Derivative of a trace with Kronecker product

Say I have the following expression: $f=\mathbf{Tr} (S^T L S)$,where $S=(B\bigotimes A)H$ Then what's the derivative of $f$ with respect to $A$ and $B$ ?