Questions tagged [vectorization]

Question 1

Let $X \in M_n(\mathbb{R})$ be a matrix and $A \in M_n(\mathbb{R})$ be a matrix (not necessarily invertible) with all entries strictly positive such that $A X A^\top=X$. Prove or disprove $X=O_n$. If ...

Question 2

Let $\bf A$ be an invertible $m \times m$ real matrix. Is there any simplified expression for the following $m^2 \times m^2$ rank-$1$ matrix? $$\operatorname{vec}({\bf A}) \operatorname{vec}^\top \...

Question 3

My question can be summarised as: Is the convex hull of a set of matrices identical to the convex hull of the vectorised set mapped back into a set of matrices? The very obvious answer seems yes due ...

Question 4

Let $X\in{\mathbb{R}}^{m{\times}n}$ be a general matrix, what's the derivative of $X{\otimes}X$? i.e. I want to calculate $\frac{{\mathrm{d}}(X{\otimes}X)}{{\mathrm{d}}X}$, or the derivative of its ...

Question 5

I have a symmetric positive definite matrix $L\in\mathbb{R}^{n\times n}$ and its Cholesky factor $G\in\mathbb{R}^n$ such that $L=GG^T$. Called $\mathrm{vec}:\mathbb{R}^{n\times n}\to\mathbb{R}^{n^2}$ ...

Question 6

I've been trying to generalize a vector-matrix-vector product that represents the $i$th element of a vector $v$, but I can't figure out how to put it into a concise form. Let $v \in \mathbb{R}^{n}$ be ...

Question 7

Below are examples of the matrix I have in mind: $$ P_{2} = \begin{bmatrix} 1 & & & \\ & & 1 & \\ & 1 & & \\ & & & 1 \end{...

Question 8

Given a function, that accepts a matrix and returns a vector: $g(X):\mathbb{R}^{nxm}\rightarrow \mathbb{R}^{m}$, s.t. $g^{(i)} = t\cdot X^{(i)}$; where $X$ is an $n\times m$ matrix, $t$ is a row ...

Question 9

Suppose we have the following matrices defined over the field of complex numbers ($\Bbb C$): a square input matrix $\mathbf{U}$ with dimensions $n \times n$ a symmetric convolution kernel $\mathbf{H}...

Question 10

I am looking for a more elegant way to confirm the following intuition: Assume that $A$ and $B$ are two square $p\times p$ matrices. It seems there should always be some matrix $C$ such that $\...

Question 11

I want to find solutions for matrices $A\in \mathbb{R}^{m\times n}$ and $B\in \mathbb{R}^{n\times m}$ in the following equations: $$ \left\{ \begin{matrix} A^TM_1 = A^T(AB\odot M_2) \\ ...

Question 12

I have a constraint optimization problem formulated in a diagonal matrix form: $ P_3:~ min_{x} \quad \|A X(t) - Y(t)\|^2 \\ \text{subject to} \quad X^*(t) \cdot X(t) = \mathbb{I} $ I need to ...

Question 13

Here is the equation I want to solve: $$\frac{d\vec{A}}{d\vec{C}}=\frac{d\mathbf{M}(\vec{C})\vec{B}}{d\vec{C}}$$ where $\vec{A} = \mathbf{M}(\vec{C})\vec{B}$ $\vec{B}$ is a constant vector $\vec{C}$ ...

Question 14

Basically the title. For $A \in \mathbb{R}^{n \times n}$, how does $\text{rank}(A)$ relate to $\text{rank}(\text{Diag}(\text{Vec}(A)))$ where $\text{Diag}(\text{Vec}(A) \in \mathbb{R}^{n^2 \times n^2}...

Question 15

This question is hard to ask, but I'll try to be as specific as I can. For matrices $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{p \times q}$, is there an equivalent representation (see below) ...

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Questions tagged [vectorization]

Proving or disproving the existence of a matrix

Simplifying $\operatorname{vec}({\bf A}) \operatorname{vec}^\top \left({\bf A}^{-1}\right)$

Convex hull of matrices equal to convex hull of their vectorised equivalent?

What's the derivative of the Kronecker product of a matrix and itself? [closed]

Derivative of the Cholesky factor

How can I generalize a vector-matrix-vector product into a vector?

Does this permutation matrix have a name?

How to vectorize a function that accepts a matrix and returns a vector in Matlab/octave/python

Can the transpose of cropped block-Toeplitz matrix be represented as a cropped 2D convolution?

$\operatorname{vec}(AB) = \operatorname{vec}(A) \otimes \operatorname{vec}(B)$

Solve matrix equations involving vectorization and Kronecker product

Partial derivative of complex matrix products

How to differentiate a vector related to a matrix w.r.t. a vector by using Frobenius product notation

How does $\text{rank}(A)$ relate to $\text{rank}(\text{Diag}(\text{Vec}(A)))$?

Equivalent form of vectorization of Kronecker product

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