Questions tagged [hadamard-product]

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

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How do you calculate the degree of a polynomial involving Hadamard products?

If I have the following polynomial with an indeterminate vector $\vec{x}$, how do I compute the degree of let's say $\vec{x} \cdot \vec{x}$? (with $\cdot$ being the Hadamard product)
David 天宇 Wong's user avatar
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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Is interlacing of zeroes of parabolas preserved by Hadamard power?

Knowing that two parabolas in general form: $$f(x)=x^2+ax+b \qquad g(x)=x^2+cx+d$$ have real,interlacing zeroes (i.e. $x_1<y_1<x_2<y_2$), is it true that $\forall p>1, p\in \mathbb{R}$ $$f(...
michelle's user avatar
1 vote
1 answer
81 views

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) \\ ...
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Generalized eigenvalue problem with real symmetric matrices and Hadamard product

I have a generalized eigenvalue problem of the form $$ (S \circ A) v = \lambda S v $$ where $\circ$ denotes the elementwise or Hadamard product and $S$ and $A$ are real symmetric matrices As far as I ...
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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
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Derivative of determinant of matrix product of matrix and Hadamard product of 2 matrices

My problem is how to compute the partial derivative of $ f( \boldsymbol{\Lambda}_{\mathsf{h}}, \boldsymbol{\Lambda}_{\mathsf{x}}, \boldsymbol{\Lambda}_{\mathsf{z}} ) $ with respect to $ \boldsymbol{...
Prey.Q C's user avatar
1 vote
1 answer
57 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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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
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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
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73 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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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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Hadamard product of two sets of vectors?

For any two vectors $x,y$ in $\mathbb{R}^N$, the Hadamard product (element-wise multiplication) is defined by $$x\circ y\equiv(x_1y_1,\cdots,x_Ny_N)\in\mathbb{R}^N.$$ I am curious if anyone has ...
Yi-Hsuan Lin's user avatar
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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26 views

Determinant of Hadamard power of an all-positive matrix

Suppose $C$ is $n\times n$ positive definite matrix with $C_{ii}=1$ and off-diagonals between zero and one $0<C_{ij}<1$. what can be said of determinant of $C^{\odot k}$ in this special case? In ...
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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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$n$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
24 views

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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112 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 $$v^T (A \circ B) v = v^TAv \cdot v^TBv$$ ...
a6623'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
63 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 ...
Hans's user avatar
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2 votes
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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),...
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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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-2 votes
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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
56 views

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 $...
lyh458's user avatar
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1 vote
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System of second-degree polynomial equations ("Hadamard vector polynomial equation")

I am interested in solutions of the following system of polynomial equations that can be written as an 'element-wise' vector polynomial: $$ \mathbf{0} = \mathbf{A}^{(0)}\mathbf{1} + \mathbf{A}^{(1)}\...
qubical's user avatar
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What set of matrices $X,Y$ satisfy $X \circ Y = X P Y$ for at least one permutation matrix $P$?

Suppose we have three $n \times n$ matrices $X,Y,P$ where $X,Y \in \mathbb{R^{n \times n}}$ and $P$ is a permutation matrix. Let us take $\circ$ to be the element-wise product between two matrices. ...
Galen's user avatar
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"Leave-on-out Correlation" between Matrices

I'd like to enforce a special constraint in my optimization problem. The solution to my problem is a set of matrices $Q_1, ..., Q_N \in \mathbb{R}^{G \times K}$ and I'd like to make sure that: For ...
N8_Coder's user avatar
2 votes
3 answers
95 views

Derivative of Hadamard Product of two vectors

How can I compute the following derivative? $$\frac{\partial(K u \circ T u)}{\partial u}$$ $K$ and $T$ are constant matrices, $u$ is an unknown vector. and $\circ$ is Hadamard product. my solution: $$\...
Amir's user avatar
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1 answer
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Computation of two Jacobians

Definitions Consider the following function $f:\mathbb{R}^N\mapsto\mathbb{R^2}$ \begin{equation*}f(\ell_{k+1})= \left[\begin{array}{c} v_{k+1}\,\cos(h_{k+1})\\ v_{k+1}\,\sin(h_{k+1}) \end{array}\...
matteogost's user avatar
2 votes
1 answer
324 views

Inverse of matrix with hadamard product

Let $A$ and $B$, $X$ be matrices with $\mathbb{R}^{n \times n}$ where $A$, $B$ are a dense and sparse matrix, i.e., the almost elements of $B$ are zeros, respectively. I'm looking for a way to solve ...
Wanny's user avatar
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3 votes
1 answer
381 views

Lower bound on smallest eigenvalue of hadamard product of two Hermitian matrices

Let A and B be $n \times n$ real symmetric matrix. Suppose A is positive definite and denote its smallest eigenvalue as $\lambda_{\min}(A)>0$. All elements of B are positive and bounded, i.e. $0\le ...
Woody's user avatar
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2 answers
174 views

Matrix operation to exponentiate each element in a vector

I am using the following matrix algebra to obtain a vector, however, I eventually need all the resulting elements to be exponentiated. \begin{equation} \begin{split} \boldsymbol{\beta}^{\textsf{T}}\...
user avatar
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0 answers
87 views

Conditional convergence of the Hadamard product

The product $$\prod_\rho \left(1-\frac{s}{\rho}\right)$$ where $\rho$ ranges over all zeros of the Riemann zeta function is often referred to as the Hadamard product (because its convergence was first ...
Valerio's user avatar
  • 360
1 vote
1 answer
164 views

Upper-bound for nuclear norm of $A \circ (v \otimes v)$ in terms of operator norm (or nuclear norm) of matrix $A$ and $L_\infty$-norm of vector $v$.

Let $A \in \mathbb R^{n \times }$ be a psd matrix such that $\|A\|_{op} \le r_1$ and $\|A\|_{*} \le r_2$. Let $v \in \mathbb R^n$ such that $\|v\|_\infty \le r_3$. Let $B:=A \circ V$ be the Hadamard ...
dohmatob's user avatar
  • 9,140
0 votes
1 answer
294 views

Chain rule and derivative with matrix product?

I'm trying to compute some derivatives with given vectors and functions: column vector $X=[x_1,x_2,\dots,x_n ]^T$ and $Z=[z_1,z_2,\dots,z_n ]^T$, row vector $Y=[y_1,y_2,\dots,y_n ]$ $f(X,Y)=e^{XY}$ $$...
KaT7's user avatar
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2 votes
1 answer
174 views

Adjoint of Hadamard Product

0I have a linear map $f:\mathbb{R}^{n\times m}\to \mathbb{R}^{n\times m}$ defined as $$ f(X) = A \odot X \qquad \qquad \text{ for some } A\in\mathbb{R}^{n\times m} $$ where $\odot$ is the Hadamard ...
Physics_Student's user avatar
1 vote
1 answer
58 views

Determining if matrix is positive semi-definite

If $M, N \in \mathbb{R}^{n \times n}$ are symmetric and positive semi-definite then is $M^2 \circ N^2 - (M \circ N)^2 = (MM) \circ(NN) - (M \circ N)^2$ symmetric and positive semi-definite? Here $\...
El Dorado's user avatar
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0 answers
48 views

(Schatten) 1 to 1 norm of Schur multiplier with bounded coefficients.

Suppose that I have an infinite matrix $K(i,j)$ with the promise that there is some $C$ such that $\vert K(i,j) \vert \leq C$ for all $i,j \in \mathbb{Z}$. Consider the Hilbert space $l^2( \mathbb{Z})...
Frederik Ravn Klausen's user avatar
1 vote
1 answer
152 views

Solving Quadratic Matrix Equation involving Hadamard/Element-wise Product?

I have the following equation: $$x^T M x = (x \circ b)^T P (x \circ b) $$ where $x, b \in \mathbb{R}^D$ are vectors $M, P \in \mathbb{R}^{D \times D}$ are matrices $b, P$ are known $\circ$ denotes ...
Rylan Schaeffer's user avatar
5 votes
1 answer
1k views

Notation for sum over element wise multiplication

Im looking for a typical notation for the sum over the elements after an element-wise multiplication of two matrices $A$, $B$ (hadamard product). Is it correct to write $\sum A \odot B$ ...
Ai4l2s's user avatar
  • 151
0 votes
1 answer
202 views

Hadamard Mixed-Product Expression for nxn Matrix and nx1 Vector Terms?

I have a problem about a Hadamard product (i.e. elementwise multiplication) between two terms, each term being the matrix dot product between an $n\times n$ sized matrix $M$ and the $n\times 1$ vector ...
Pietakio's user avatar
3 votes
0 answers
136 views

Generating function for the squared Catalan numbers

The squares of the Catalan numbers: 1, 1, 4, 25, 196, 1764... are given in OEIS A001246. In the OEIS entry two ordinary generating functions for the series are given in terms of elliptic integrals/...
maxwelldecoherence's user avatar
1 vote
0 answers
35 views

Submultiplicativity of Hadamard product for nonnegative

Let $A\in {\mathbb R}^{m\times n}_+$ be matrix with positive entries, and $B\in {\mathbb R}^{m\times n}$. Is it true: $$ \sigma_1(A\circ B)\leq \max_{i,j}a_{ij} \cdot \sigma_1(B) $$ In fact, I ...
qwerty43's user avatar
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0 votes
1 answer
127 views

Inequality for Hadamard product of matrices

I have nonnegative matrices $A,B,C$ such that $A = B \circ C$. It seems that $A^N \leq B^N \circ C^N$ (i.e. $(B\circ C)^N \leq B^N \circ C^N$) is true element-wise. I would like to show that this it ...
Jacob A's user avatar
  • 575
0 votes
1 answer
85 views

$\sum(F\circ F)\geq x_1^2+\cdots+x_n^2$

Let $A,B$ be $n\times n$ matrix, denote by $A\circ B=(a_{ij}b_{ij})$. Let $C$ be an invertible real matrix. $D=diag(x_1,\cdots,x_n)$, $F=CDC^{-1}$. Show that $\sum(F\circ F)\geq x_1^2+\cdots+x_n^2$, ...
xldd's user avatar
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2 votes
2 answers
91 views

Proof that $\|\vec{x} \odot \vec{y}\| \leq \|\vec{x}\|\|\vec{y}\|$

I suspect that $$\|\vec{x} \odot \vec{y}\| \leq \|\vec{x}\|\|\vec{y}\|,$$ where $\vec{x}, \vec{y} \in \mathbb{R}^n$ and $\odot$ is the Hadamard product. I have completed Monte Carlo simulations where ...
Galen's user avatar
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1 vote
2 answers
179 views

Solving matrix equation with element-wise products

I am wondering if there is a way to solve this equation for a: $$(as^T ⊙ b)n = t.$$ where: ⊙ is element-wise multiplication a is an unknown v x 1 vector s is an i x 1 $\vec{1}$ vector b is a ...
Max's user avatar
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