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)
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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 ...
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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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### 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 ...
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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 ...
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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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### 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. ...
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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 ...
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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: $$\... 0 votes 1 answer 42 views ### Computation of two Jacobians Definitions Consider the following function f:\mathbb{R}^N\mapsto\mathbb{R^2} \begin{equation*}f(\ell_{k+1})= \left[\begin{array}{c} v_{k+1}\,\cos(h_{k+1})\\ v_{k+1}\,\sin(h_{k+1}) \end{array}\... 2 votes 1 answer 324 views ### Inverse of matrix with hadamard product Let A and B, X be matrices with \mathbb{R}^{n \times n} where A, B are a dense and sparse matrix, i.e., the almost elements of B are zeros, respectively. I'm looking for a way to solve ... 3 votes 1 answer 381 views ### Lower bound on smallest eigenvalue of hadamard product of two Hermitian matrices Let A and B be n \times n real symmetric matrix. Suppose A is positive definite and denote its smallest eigenvalue as \lambda_{\min}(A)>0. All elements of B are positive and bounded, i.e. 0\le ... 0 votes 2 answers 174 views ### Matrix operation to exponentiate each element in a vector I am using the following matrix algebra to obtain a vector, however, I eventually need all the resulting elements to be exponentiated. \begin{equation} \begin{split} \boldsymbol{\beta}^{\textsf{T}}\... 0 votes 0 answers 87 views ### Conditional convergence of the Hadamard product The product$$\prod_\rho \left(1-\frac{s}{\rho}\right)$$where \rho ranges over all zeros of the Riemann zeta function is often referred to as the Hadamard product (because its convergence was first ... 1 vote 1 answer 164 views ### Upper-bound for nuclear norm of A \circ (v \otimes v) in terms of operator norm (or nuclear norm) of matrix A and L_\infty-norm of vector v. Let A \in \mathbb R^{n \times } be a psd matrix such that \|A\|_{op} \le r_1 and \|A\|_{*} \le r_2. Let v \in \mathbb R^n such that \|v\|_\infty \le r_3. Let B:=A \circ V be the Hadamard ... 0 votes 1 answer 294 views ### Chain rule and derivative with matrix product? I'm trying to compute some derivatives with given vectors and functions: column vector X=[x_1,x_2,\dots,x_n ]^T and Z=[z_1,z_2,\dots,z_n ]^T, row vector Y=[y_1,y_2,\dots,y_n ] f(X,Y)=e^{XY}$$...
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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 ...
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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 $\... 0 votes 0 answers 48 views ### (Schatten) 1 to 1 norm of Schur multiplier with bounded coefficients. Suppose that I have an infinite matrix$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})...
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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 ...
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### 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 $\sum A \odot B$ ...
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### 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 ...
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### 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/...
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### 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 ...
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### 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 ...
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### $\sum(F\circ F)\geq x_1^2+\cdots+x_n^2$

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