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Questions tagged [vectorization]

The vectorization of a matrix is a linear transformation that converts the matrix into a column vector.

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Any body knows the name of the product $A\circ B$, It is not Hadamard tensor product

I have seen this operation in a vectorization operation of $D=AX^{T}B$ i.e. $vec(D)=\left( A\circ B\right) vec(X),$ \begin{equation*} A\circ B= \begin{pmatrix} A_{1}B_{1}^{T} & A_{2}B_{1}^{T} &...
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20 views

Vectorized form of a Graph's Degree matrix?

A Graph's diagonal Degree matrix, $D$ is typically defined as $$ D_{ii} = \sum_{j} A_{ij} $$ i.e., each i-th diagonal of the Degree matrix is the sum of the i-th row of matrix $A$ (the Adjacency ...
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39 views

Derivative with respect to vectorized inverse Kronecker product

I am trying to derive the gradient of a function I wish to optimize, and wish to obtain the following derivative: $$ \frac{\partial}{\partial \pmb{x}} \left(\pmb{I} - \pmb{X} \otimes \pmb{X} \right)^{-...
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1answer
34 views

Derivative of vectorized block matrix in terms of derivatives of vectorized blocks

Suppose I have some block matrix $\pmb{Y}$ that is a function of $\pmb{x}$: $$ \pmb{Y} = \begin{bmatrix} \pmb{A} & \pmb{C} & \pmb{E} \\ \pmb{B} & \pmb{D} & \pmb{F} \\ \end{bmatrix}. $$ ...
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1answer
39 views

Derivative of matrix w.r.t. its own vectorized version

I am unable to find what would be the derivative of a $m \times m$ real matrix $A$ with respect to $(\mathrm{vec}(A))^T$ (where $T$ is transpose and $\mathrm{vec}$ stacks the columns) without using ...
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48 views

Vectorization identity: Weyl matrices

Take a ring $Z_n = \{0,1,...,n-1\}$. All operations occur modulo $n$. Consider the $n^{th}$ roots of unity $\omega_n = \exp\left(\frac{2\pi i}{n}\right)$. Now define $$X = \sum_a \vert{a+1}\rangle\...
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What is the vectorized form of this expression?

I want to figure out a general formula for vectorizing computations. For example, if I am given matrices, A,B such that A is $m \times q$, B is $n \times q$. And I want to compute the matrix $X$, ...
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33 views

Vectorization identity proof

I'm trying to prove the identity $\vert vec(AXB)\rangle = A\otimes B^T \vert vec(X)\rangle$, where $\vert vec(L)\rangle := \sum_{ij} L_{ij}\vert i\rangle\vert j\rangle$ for any $L:= \sum_{ij}L_{ij}\...
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15 views

Vectorization of a value matrix with a position vector in order to build a target matrix T

I have a matrix $A$ of the size $k \times N$ where the first value denotes the number of rows and the second the number of columns. Moreover, I have a vector $p$ which contains numbers between $0$ and ...
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1answer
32 views

The location of the diagonal elements in the half-vectorization of a matrix

I'm trying to find a formula for the index of the diagonal entry when organized in a vector given by the half-vectorization operator. Assume that we have a symmetric matrix $$ M = \begin{bmatrix}m_{...
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38 views

Cost function - vectorized implementation

I have a problem regarding how to vectorize, more specifically the problem below: Repeat { $$\theta_j := \theta_j - \frac{\alpha}{m} \sum\limits_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})x_j^{(i)}$$ } ...
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1answer
56 views

vectorization of matrices

$vec(\boldsymbol{\beta}\boldsymbol{\sigma}_{n-1}^2\boldsymbol{e}_j'\boldsymbol{\sigma}_{n-1}^2\boldsymbol{\alpha}_1^\prime)=((\boldsymbol{\alpha}_1\boldsymbol{e}_j')\otimes\boldsymbol{\beta})vec(\...
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Why are stochastic processes governing the embedding of separable Hilbert spaces into a common non-separable Hilbert space?

By starting from a modeling platform in which multiple separable Hilbert spaces that are defined on top of the same vector space float over a infinite dimensional background separable Hilbert space ...
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How do I use vectorization to simplify matrix integration problem?

Can someone show the detailed procedures for proof: \begin{equation*} \text{vec}\left(\int^T_0ds\,e^{-Ks}\Sigma\Sigma^\text{T}e^{-K^\text{T}s}\right) = \left(K\otimes I+I\otimes K\right)^{-1}\text{...
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1answer
568 views

Vectorization and transpose: how are $\text{vec}(W^T)$ and $\text{vec}(W)$ related?

In solving for a gradient, I ended up with a differential that looks similar to: $$ dT = (a^T \otimes b^T)\ \text{vec}[d[W]^T] + (b^T \otimes c^T)\ \text{vec}[d[W]] $$ and I am trying to solve for $\...
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72 views

Solve vectorial equations

I'm trying to figure out a passage for reducing a vectorial equations ! for doing this somebody told me to use a program of symbolic calculation (matlab,maple, mathematical .. or python as well .. ) I ...
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Vectorization of expressions: How to develop a visual intution

I'm currently analyzing the expressions for backpropagation in machine learning and it takes me a lot of time to convert my derived formulas into matrices. I can't find anything helpful on google, ...
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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
73 views

Derivative eigensystem

I have a diabatic potential energy matrix, $V(r,Z)$, (real symmetric) for a 2-level system with two nuclear coordinates, $(r,Z)$: $$ V(r,Z)= \begin{pmatrix} V_{11}(r,Z) & V_{12}(r,Z)\\ V_{12}(r,...
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What is the term for the result of an Euler vector?

An Euler 'vector' is not a vector, because it has direction, but no magnitude. It is also not a scalar. So, what is an Euler vector? A direction?
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1answer
313 views

Vectorize 3D discrete Fourier transform

I'm trying to express the $N$-dimensional discrete Fourier transform (DFT) of an $N$-dimensional array as the product between a matrix and a vector. If $N=2$ the problem is quite simple: given a $n\...
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1answer
35 views

Vector Parametric Form Flat

Please see the image with the question and my answer. The thing I'm having an issue with is understanding what this vector description actually is. Is it a 2-dimensional flat in $\mathbb{R}^6$? In ...
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1answer
98 views

How can I vectorize this sum (Hessian)?

Let $X$ be a $n \times p$ matrix, $n>p$. I derived a (Hessian) matrix as follows: $$H(\theta) = \sum_{i=1}^n g(x_i,\theta) x_i x_i^T$$ where $g(x_i,\theta)$ is a scalar function and $x_i$ is the $...
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Why use the Kronecker product?

I have found many references on Kronecker product but I did not see any reference talking about why this way of multiplication exist and whats the intuitive use of this particular product. Appreciate ...
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1answer
143 views

Differentiation w.r.t. the $\mbox{vec}$ operator

I am stuck at solving the following derivative $$\frac{d \mbox{vec} (X^T X)}{d \mbox{vec} (X)}$$ where $X$ is an $m \times n$ matrix and $\mbox{vec}$ is the vector/stack operator. I have tried using ...
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1answer
626 views

Inverse Vectorization Vec^-1 [duplicate]

Hope that you will find this post in good health. I am Mr. Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using. During ...
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0answers
330 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 $\...
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95 views

Vectorization of the nonzero entries of a matrix

I am familiar with the $\textrm{vec}(A)$ operator, where the columns of a matrix $A$ are stacked into a vector. If my matrix $A$ has zero-entries, is there a standard notation/operator like $\textrm{...
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What is the inverse of the $\mbox{vec}$ operator?

There is a well known vectorization operator $\mbox{vec}$ in matrix analysis. I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the ...