# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$V=\left[\begin{... 0answers 112 views ### Product of matrix-valued normal densities and Kronecker product I am trying to find an expression for the mean, column-covariance and row-covariance matrices of the product of two matrix-valued Normal distributions. Here is what I've tried in a special case I ... 0answers 334 views ### Writing an expression in terms of vectorization operator \mbox{vec} (X) I am new with vectorization and Kronecker products. I need to write the scalar value$$\mathbf{a}^{T}\mathbf{X}\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{a}$$in terms of \... 0answers 136 views ### Efficient evaluation of quadratic form for a Kronecker-decomposable covariance matrix? Well, it's in the title. I'd like to evaluate \mathbf{y}'K^{-1}\mathbf{y} in the case that K is decomposable into a Kronecker product (e.g. K = A\otimes B\otimes C). (EDIT: K is also symmetric ... 0answers 412 views ### Kronecker delta representation of a matrix (Quantum raising / lowering operators) The Kronecker Delta is commonly used to represent a diagonal matrix:$$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$... 0answers 67 views ### Loop integral with Kronecker-delta notation I worked through a problem and arrived at the final solution (which is correct), however, one part of it should equal zero mathematically. This is the part that should equal zero: F_{2i}= (\frac{... 0answers 86 views ### Quadratic form of Kroenecker products of skew-symmetric matrices I am trying to understand under which conditions on P=P^\top>0 , C=C^\top, the following quadratic form is zero:$$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$... 0answers 105 views ### an inequality about kronecker product with eigenvalues question Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of 0=\lambda_{1}(L)<\lambda_{2}(L)\leq...\leq\... 0answers 8 views ### Taylor expansion using Kronecker tensor I have the following function (just used as an example): y_t=g(y_{t-1},\epsilon_t, \sigma) of which I have the following second-order Taylor expansion around a point such that y=\bar{y}, \ \... 0answers 45 views ### Property on Kronecker product I read a paper and there was an equation which was finally derived an equivalent expression as$$ L = L_{T} \otimes I_{G} + I_{T} \otimes L_{G} = {\color{blue}{L_{T} \times L_{G}}} , $$and ... 0answers 35 views ### Question about creating a 3-rank tensor in numpy with python3 I'm trying to create a 3-rank tensor in numpy, with python3.x. I need to create this 3-rank tensor A in a very particular way. If I have 3 matrices, let's say that all of them are some Y matrix (... 0answers 18 views ### A question about spectral norm and kronecker product Assume A is a n\times n matrix and B=\left[\begin{matrix}B_{11}&B_{12}\\B_{21}&B_{22}\\\end{matrix}\right], then do we have$$\|A\otimes B\|=\|\left[\begin{matrix}A\otimes B_{11}&A\...
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 ...