Questions tagged [vectorization]
The vectorization of a matrix is a linear transformation that converts the matrix into a column vector.
0
votes
0answers
24 views
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} &...
0
votes
1answer
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 ...
1
vote
1answer
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:
$$
\frac{\partial}{\partial \pmb{x}} \left(\pmb{I} - \pmb{X} \otimes \pmb{X} \right)^{-...
0
votes
1answer
34 views
Derivative of vectorized block matrix in terms of derivatives of vectorized blocks
Suppose I have some block matrix $\pmb{Y}$ that is a function of $\pmb{x}$:
$$
\pmb{Y} = \begin{bmatrix}
\pmb{A} & \pmb{C} & \pmb{E} \\
\pmb{B} & \pmb{D} & \pmb{F} \\
\end{bmatrix}.
$$
...
1
vote
1answer
39 views
Derivative of matrix w.r.t. its own vectorized version
I am unable to find what would be the derivative of a $m \times m$ real matrix $A$ with respect to $(\mathrm{vec}(A))^T$ (where $T$ is transpose and $\mathrm{vec}$ stacks the columns) without using ...
0
votes
0answers
48 views
Vectorization identity: Weyl matrices
Take a ring $Z_n = \{0,1,...,n-1\}$. All operations occur modulo $n$. Consider the $n^{th}$ roots of unity $\omega_n = \exp\left(\frac{2\pi i}{n}\right)$. Now define
$$X = \sum_a \vert{a+1}\rangle\...
0
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0answers
29 views
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 $X$, ...
0
votes
0answers
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}\...
0
votes
0answers
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 ...
-1
votes
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
$$ M = \begin{bmatrix}m_{...
0
votes
0answers
38 views
Cost function - vectorized implementation
I have a problem regarding how to vectorize, more specifically the problem below:
Repeat {
$$\theta_j := \theta_j - \frac{\alpha}{m} \sum\limits_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})x_j^{(i)}$$
}
...
0
votes
1answer
56 views
vectorization of matrices
$vec(\boldsymbol{\beta}\boldsymbol{\sigma}_{n-1}^2\boldsymbol{e}_j'\boldsymbol{\sigma}_{n-1}^2\boldsymbol{\alpha}_1^\prime)=((\boldsymbol{\alpha}_1\boldsymbol{e}_j')\otimes\boldsymbol{\beta})vec(\...
0
votes
0answers
27 views
Why are stochastic processes governing the embedding of separable Hilbert spaces into a common non-separable Hilbert space?
By starting from a modeling platform in which multiple separable Hilbert spaces that are defined on top of the same vector space float over a infinite dimensional background separable Hilbert space ...
-1
votes
1answer
30 views
How do I use vectorization to simplify matrix integration problem?
Can someone show the detailed procedures for proof:
\begin{equation*}
\text{vec}\left(\int^T_0ds\,e^{-Ks}\Sigma\Sigma^\text{T}e^{-K^\text{T}s}\right) = \left(K\otimes I+I\otimes K\right)^{-1}\text{...
1
vote
1answer
568 views
Vectorization and transpose: how are $\text{vec}(W^T)$ and $\text{vec}(W)$ related?
In solving for a gradient, I ended up with a differential that looks similar to:
$$
dT = (a^T \otimes b^T)\ \text{vec}[d[W]^T] + (b^T \otimes c^T)\ \text{vec}[d[W]]
$$
and I am trying to solve for $\...
0
votes
2answers
72 views
Solve vectorial equations
I'm trying to figure out a passage for reducing a vectorial equations ! for doing this somebody told me to use a program of symbolic calculation (matlab,maple, mathematical .. or python as well .. ) I ...
1
vote
0answers
114 views
Vectorization of expressions: How to develop a visual intution
I'm currently analyzing the expressions for backpropagation in machine learning and it takes me a lot of time to convert my derived formulas into matrices. I can't find anything helpful on google, ...
2
votes
1answer
123 views
Finding Hessian of tr ((AB)' (AB))
I'm trying to find Hessian of $\text{tr}((AB)' (AB))$ where $A,B$ are matrices. There are nice expressions for $H_{AA}$ and $H_{BB}$ using standard approach from Magnus 1 , can anyone suggest how to ...
1
vote
1answer
73 views
Derivative eigensystem
I have a diabatic potential energy matrix, $V(r,Z)$, (real symmetric) for a 2-level system with two nuclear coordinates, $(r,Z)$:
$$
V(r,Z)=
\begin{pmatrix}
V_{11}(r,Z) & V_{12}(r,Z)\\
V_{12}(r,...
0
votes
0answers
32 views
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?
0
votes
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\...
1
vote
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 ...
0
votes
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 $...
7
votes
2answers
3k views
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 $\...
3
votes
0answers
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{...
9
votes
2answers
2k views
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 ...
