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Questions tagged [hadamard-product]

For questions about Hadamard product between two matrices, or it can concern analytic functions.

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Full Rank of Hadamard product matrix

Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$: $$ C:=\left[\begin{array}{cccc} c_1^1 & c_2^1 & \cdots & c_n^1 \...
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Eigenvalues of Composition of Hadamard Operations of Low Rank Matrices

I am interested in the eigenvalues of $$ee^T \oslash (aa^T - a^{\odot2}(a^{\odot2})^T )^{\odot \frac{1}{2}},$$ where $a \in \mathbb{R}^n$ and $e$ is the vector with all entries equal to one. Can we ...
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Is the Hadamard Product of two laplacian operators allowed to get some kind of biharmonic operator?

I'm currently working on my masters thesis in computer science and from this point I'm not that into this subject. Right know I try to understand the steps the authors of this paper did to get the ...
dontoronto's user avatar
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Hadamard product: derivatives

How to compute the partial derivatives of the following expression $$ \phi(x,y)=Ax\circ By, $$ with the Hadamard product $\circ$, where $A(n-2,n)$, $B(n-2,n)$ are the matrices $$ A=\left[\begin{array}{...
justik's user avatar
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Quadratic form with Hadamard product, minimization

Let $x,y,a,b$ be the column vectors $(n,1)$ and $D_{1}(n-1,n),D_{2}(n-1,n)$ be the central difference matrices $$ D_{1}=\left[\begin{array}{ccccccc} -0.5 & 0 & 0.5 & \cdots & 0 & 0 ...
justik's user avatar
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Rank of Hadamard product with rank 1 matrix? [closed]

Let $\circ$ be the Hadamard product (i.e. element-wise multplication). We know that $$ \operatorname{rank}(A\circ B) \leq \operatorname{rank}(A)\operatorname{rank}(B) $$ If we are given that $A$ is a ...
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Deduce $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix}$ or the simple parity-check matrix just from the final matrix-vector product equations

A linear combination like $A_{1}+ A_{2}$ serves as a backup of $A_{1}$ when $A_{2}$ is known, and serves as a backup of $A_{2}$ when $A_{1}$ is known. As a result of linearity, any two out of $A_{1}x$,...
triple_tactic's user avatar
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Using the r-Lambert function to solve a system of transcendental equations

I am trying to use the r-Lambert function applied to a vector in order to solve a system of transcendental equations, however, I am facing some difficulties when trying to obtain the right expression ...
Ignacio Canabal's user avatar
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Derivative of real-valued function that takes a matrix

I want to compute the partial derivative of a real-valued function that takes matrices as argmuents. The function has the form $$F(x,y,z) = ||g(x) \odot (S \cdot y) - z||,$$ where $x, y, z, S \in \...
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Hadamard products and sums of reciprocals of solutions to $x=\tan x$

There are infinitely many real solutions to the equation $\tan x=x$. Denote the increasing sequence of positive solutions by $ (\lambda_n)_{n=1}^{\infty}$. I want to evaluate the sum $$ \sum_{n=1}^{\...
Dave's user avatar
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What is the derivative of $\text{diag}(x)^2 Ax$?

Let $\mathbf{x}$ be a column vector of $\mathbb{R}^n$. Now, if $$ f(x) = \text{diag}(x)^2 Ax$$ what is the derivative of $f$ with respect to $x$? For now, I looked at different formulations for the $\...
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How can we ensure the matrix $\mathbf{T} - ( \mathbf{C} \circ \mathbf{A} )( \mathbf{C} \circ \mathbf{A} )^H$ is positive semi-definite(PSD)

Specifically, the operator $\circ$ denotes the hadamard product, the matrix $\mathbf{T}$ is a low rank toeplitz PSD matrix , the matrix $\mathbf{A}$ is a matrix in vandermonde structure and its ...
an chen's user avatar
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What does it say about two curves $\vec x (t)$ and $\vec y (t)$ if $|| \vec x(t) \cdot \vec y(t) || \leq || \vec x(t) \odot \vec y(t) ||$?

What does it say about two curves $\vec x (t)$ and $\vec y (t)$ in $\mathbb{R}^n$ if $$|| \vec x(t) \cdot \vec y(t) || \leq || \vec x(t) \odot \vec y(t) ||?$$ That is, what are some necessary and non-...
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Solving linear equation with quadratic hadamard product

I have the following equation that I can solve iteratively, but I was wondering if it has an analytical solution that can speed up my computation. $$ x = B(x \odot x \odot b) + a $$ where $x, a, b \...
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Hadamard's theorem proof related steps

$$ \int_0^{2 \pi} \frac{\rho e^{i \theta} d \theta}{\left(\rho e^{i \theta}-z\right)^{h+2}}=\frac{-i}{h+1}\left[\frac{1}{\left(\rho e^{i \theta}-z\right)^{h+1}}\right]_0^{2 \pi}=0 $$ $\therefore$ We ...
Nothing's user avatar
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Definition of Hadamard product between matrix and vector

I have read here that: $$ M \circ \vec{v} = \operatorname{Diag}(\vec{v}) \, M $$ That is, the Hadamard product between an $n\times m$ matrix and an $n \times 1$ vector is equivalent to the dot product ...
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Does this matrix operation exist?

let $\mathbf{r}_p(u)=(1,u,u^2,\ldots,u^p)'\in\mathbb{R}^{1+p}$ and let $$\mathbf{R}_p = [\mathbf{r}_p(x_1), \mathbf{r}_p(x_2), \cdots, \mathbf{r}_p(x_n)]'_{n\times(1+p)}.$$ Moreover, let $\mathbf{Z}\...
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The Inner Product of a Hadamard Product

So let's say I have the inner product: $$\vec{y}_1^H \vec{y}_2 = (\vec{x}\circ\vec{h}_1)^{H} (\vec{x}\circ\vec{h}_2) = \sum_{i} (x_i^{\ast} h_{1,i}^*) (x_i h_{2,i}) = \sum_{i} |x_i|^2 h_{1,i}^* h_{...
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Inequalities about trace of Hadamard product of matrices

Denote $A\circ B$ as the Hadamard product of two matrices, that is, $$A\circ B=(a_{ij}b_{ij}).$$ Let $A$ be a $n\times n$ symmetric positive definite matrix. First, I know that $$tr(A\circ A)\leq tr(A^...
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Hadamard product as Projection of Tensor/Kronecker Product?

The tensor product of matrices $A, B$ is the block matrix $$A\otimes B = \begin{pmatrix}a_{11}B & a_{12}B&\dots\\ a_{21}B&a_{22}B&\dots\\ \vdots & \vdots&\ddots \end{pmatrix}.$$...
Mark Schultz-Wu's user avatar
1 vote
2 answers
140 views

Find the derivative of a diagonal matrix and norm

Find the derivative with respect to $X \in \mathbf{R}^{n \times p} $ of $$ \Phi(X) = \operatorname{Tr} \left( X^{\top} H(X) X \right) $$ where $H(X) := D(X) A D(X)$, where $A$ is symmetric and $$D(X) =...
Alaeddine Zahir's user avatar
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Integral involving Hadamard product with volume element [closed]

Consider the following integral: $$\int_{\mathbb{R}^{n\times m}}f(X)(H^T(A\odot dX)V)^{\wedge},$$ where $f:\mathbb{R}^{n\times m}\to \mathbb{R},$ $H\in O(n)$, $V\in O(m),$ $A$ is an $n\times m$ matrix ...
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How to solve an equation with Hadamard product

For positive definite matrices $A$ and $B$, I have the following identity: $$ C \circ [A (C \circ X)B] = C \circ Z $$ How can I solve this equation for $X$? Note: $C$ is a matrix with 1s and zeros. ...
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How to represent Frechet derivative as a matrix equation without expanding F(X+H)?

To preface the question that I actually want an answer to, I've read the paper by Nicholas J. Higham that computes the square root of a matrix via the Newton's method. He utilises the function $F(X) = ...
Robby Ram's user avatar
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Gradient of Hadamard product on Matrix

I wasn't able to find another question able to answer my case as all of them seemed be be able to use the Hadamard products relationship with the Frobenius inner product to remove the Hadamard product ...
Matthew Walsh's user avatar
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Derivative of a matrix valued function involving Hadamard product and SVD

Consider a map $f:\mathcal{V}_m(\mathbb{R}^n)\times\mathbb{R}^m\times \mathcal{O}(m)\to \mathbb{R}^{n\times m}$ given by: $$f(U,D,V)=A\circ UDV^T+UDV^T,$$ where $U^TU=I_m$, $V^TV=I_m$, $D$ is a ...
Wrik's user avatar
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SVD of element-wise matrix product

I'm trying to figure how, if ever possible, to optmise mathematically a problem, which I will describe hereafter. I read similar questions about this topic, but I ask the question nevertheless since ...
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Derivative of a trace (Graph regularization) with Hadamard product

Let us assume that $A,M,L\in \mathbb{R}^{n \times n}$. The symbolic $ \circ $ represents Hadamard product. I am trying to partial derivative the following expression: $$ F=Trace((A \circ M)L(A \circ ...
Inge Teng's user avatar
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Understanding a passage in this matrix equation with the Hadamard product

i was reading the flipout paper and i stumbled over this passage. I will summarize the main point here: $x_n$ are the inputs, $W$ is the matrix of the weights. such matrix can be decomposed as the ...
Alucard's user avatar
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4 votes
1 answer
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Series of product of legendre polynomials with shifted degree

I am working on some quantum mechanics and I would love to find a closed expression for the series $$ S(x,y) = \sum_{l=0}^\infty P_l(x) P_{l+1}(y) $$ $x,y \in \left[ -1, 1\right]$ and also for its ...
Jakub Konarek's user avatar
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Solve matrix equations involving vectorization and Kronecker product

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) \\ ...
Mokoghost's user avatar
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1 answer
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Matrix equation involving a Hadamard product?

I'm trying to find out the solution to B for the following matrix equation: $$ A^TM_1=A^T(AB\odot M_2) $$ where $A\in \mathbb{R}^{m\times n}$, $B\in \mathbb{R}^{n\times m}$, and $n<m$. $M_1, M_2\in ...
Mokoghost's user avatar
1 vote
1 answer
94 views

Correctly accounting for Hadamard product

I have an equation of the following form: $$ \frac{\partial a}{\partial t} = Da + Fe^{i\mu \theta}$$ where $a = a(\theta,t)$ is a function and $D$ is a linear operator. When discretized for solving ...
Paddy's user avatar
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1 answer
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How to prove Hadamard product rank upper bound hold?

In enter link description here, @Ben Grossmann mentioned that if one of the matrices has rank 1 and no non-zero entries, $rank(A\circ B)= rank(A)rank(B)$. My progress is as follows, where the ...
Cuz Taylor's user avatar
1 vote
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Eigenvectors of Hadamard product of a positive definite matrix and a symmetric positive semidefinite matrix

Let $b \in (0,1]$ and $n<\infty$. Consider the following square matrices: Symmetric positive semi-definite matrix $A \in M_{n \times n}(\mathbb{R})$, and $B \in M_{n \times n}(\mathbb{R})$ with $(...
Bob Edson's user avatar
4 votes
0 answers
189 views

A probabilistic proof of Oppenheim's inequality?

Oppenheim's inequality is a standard result about the Hadamard product of positive definite matrices. It goes as follows, let $A=(a_{ij})_{i,j\leq n},B=(b_{ij})_{i,j\leq n} \in S_n^{++}$ where $S_n^{++...
PAM's user avatar
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-1 votes
1 answer
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Trace inequality of Hadmard product

For any Hermitian matrix $\bf A$ and invertible matrix $\bf B$, how do I derive the inequality below $${\rm tr}(({\bf B}^H{\bf A}{\bf B}) \circ({\bf B}^H{\bf A}{\bf B}))\geq \lambda_{\min}^4({\bf B}){\...
Robin's user avatar
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1 vote
1 answer
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How to solve the problem of trace optimization which includes Hadamard product?

I have the following minimization problem, where I want to find W, \begin{align} &\min \mathrm{tr} (((W^TK)\circ(W^TK))^T((W^TK)\circ(W^TK))L)\\ &\text{s.t.} ~ W^TKHKW = I \end{align} where $\...
gouchuan's user avatar
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How to generate a 3D matrix from a 2D matrix through column-wise Hadamard product?

I have a matrix defined as, $ {\bf G}=[{\bf g}_1 \quad {\bf g}_2\quad...\quad{\bf g}_N], $ where ${\bf g}_i$ is a column vector of the length $N$. The tensor is defined as, ${\bf M}(i,j)={\bf g}_i \...
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What can one say about the eigen spectrum of the Hadamard or pointwise product of two matrices A and B

Can we give bounds or infer any type of information about the spectrum of the point-wise product of two matrices $A$ and $B$ given knowledge about the spectrum of $A$ and $B$ ? Are there non trivial ...
userrandrand's user avatar
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1 answer
154 views

$n$th-order Hadamard power (or root) of a vector

I am familiar with the notation of the $n$th-order Hadamard power $$ \mathbf{A}^{\circ n} $$ or root $$ \mathbf{A}^{\circ \frac{1}{n}} $$ I wonder if it is sensible to use this notation for vectors as ...
Rubem Pacelli's user avatar
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1 answer
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Matricial Equation with both Pointwise and Standard Multiplication

I am having trouble simplifying an equation that contains both pointwise (Hadamard) and standard matricial multiplication. Given that $I_{n\times1}$ and $K_{n\times1}$ are real $n\times1$ matrices; $...
Ricardo's user avatar
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438 views

Does matrix multiplication distribute over the Hadamard product?

I know that the Hadamard product is distributive over addition. But suppose that $A, B \in \mathbb{R}^{n \times n}$ and $v \in \mathbb{R}^n$. Then can we say $$v^T (A \circ B) v = v^TAv \cdot v^TBv$$ ...
greg115's user avatar
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Hadamard (element-wise multiplication) product rank

I am having some problems on understanding an inequality regarding the rank of the Hadamard product (element-wise product). I have $B=A\circ A$ where $A$ is a $n\times r$ matrix, and $\circ$ is the ...
Adelinne's user avatar
1 vote
2 answers
68 views

Solve system for elements of a matrix

I have a system of $n$ equations which follows a particular pattern as follows (showing the case $n=3$): $$\phi = a_1 + \psi_2 a_2 + \psi_3 a_3 \\ \phi = \psi_1 a_1 + a_2 + \psi_3 a_3\\ \phi = \psi_1 ...
Colin's user avatar
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Interpolation from unevenly distributed points of function with compactly supported Fourier transform

It is not so hard to prove the Nyquist-Shannon theorem. It basically says a real valued function $f$ having a compactly supported Fourier transform function $\hat f$ can be interpolated precisely from ...
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Is there a generalization of this formula for the determinant of the Hadmard product

Let $A$ and $B$ be any complex $2 \times 2$ matrices. Then a short calculation gives that $$ \det(A \circ B) = \frac{1}{2}\left( \det(A) \operatorname{perm}(B) + \operatorname{perm}(A) \det(B) \right),...
Malkoun's user avatar
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12 votes
1 answer
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An invertible matrix is orthogonal if and only if the inverse is equal to the transpose on nonzero elements

Let $A$ be an invertible real matrix, and suppose that $(A^{-1})_{i,j} = (A^{T})_{i,j}$ whenever $(A^{T})_{i,j}\ne 0$. Is it true that $A$ is orthogonal? I found this statement in a paper without ...
Exodd's user avatar
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What is the gradient of $x \mapsto \frac{1}{2}(x^2)^\top A(x^2)$?

Let matrix $A$ be sparse and symmetric but not semidefinite. Since I would like to use projected gradient descent, I must find the gradient of $x \mapsto \frac{1}{2}(x^2)^\top A(x^2)$, where $x^2 = \...
someone random's user avatar
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1 answer
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Derivative of Hadamard Product Multiply by Summing Vector (each element is 1)

Given $\mathbf{A} \in \mathbb{R}^{k \times n}$, diagonal matrix $\mathbf{W} \in \mathbb{R}^{k \times k}$, $$ \mathbf{F}(\mathbf{X}) = \mathbf{W}(\mathbf{AX} \odot \mathbf{AX}) \mathbf{1}_{3} $$ where $...
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