# 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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### 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 ... 58 views

### 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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### 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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### 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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Given quadratic norm: $f=\boldsymbol{g}^T\boldsymbol{p}=\boldsymbol{g}^TQ\left[\begin{array}{@{}c@{}} 0 \\ 0 \\ 1 \end{array} \right]$ where $\boldsymbol{g}=\left[\begin{array}{@{}c@{}} 0 \\ 0 \\ -9.... 2 votes 1 answer 91 views ### Extremely complex vector-matrix expression and its differentiation by vector Given:$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}{@{}...
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 ...