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

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### 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 ...
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### 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 ...
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### Order of growth of $\prod_{n=1}^{\infty}(1-a^nz)$ for $0<|a|<1$

This question is from Conway Complex Analysis, page 287, exercise 9(a). My attempt: Write the product as $\underset{n}\prod(1-\frac{z}{b^n})$, where $b=1/a$. First note that this entire function has ...
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Normally, when we have two dimensional matrices X and Y with $NxN$ dimension, we can find $A_{ni} = X_{ni} Y_{in}$ by using hadamard product, so the $ni'$th element of $X.*Y'$ will give us $A_{ni}$. ...
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### Convert Hadamard product to Matrix product (simplified)

So I have a matrix $Z \in \mathbb{R}^{m \times d}$ (which has repeated row vectors of size m) and $A \in \mathbb{R}^{m \times d}$ and I use hadamard product for them $Z \circ A$. My goal is to somehow ...
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### Derivative of squared frobenius norm of hadamard product of outer product of vector with itself and matrix w.r.t. vector

I know the title is a mouth full, and there have been many similar (and probably more complicated) questions/answers on this site, but I'm stuck on this specific problem. I am working with the ...
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### Mixed Hadamard and matrix product for backprop

This question is related to the usual backpropagation equations of machine learning, but I believe it to be mathematical in nature so this should be the right place to ask. When I'm working with non-...
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### Hadamard product and relation between eigenvalues and diagonal entries

Question: Let $A=[a_{ij}]\in M_{n\times n}(\mathbb{C})$ be diagonalizable, that is $A=S \Delta S^{-1}$ for $S\in M_{n\times n}(\mathbb{C})$ non singular and $\Delta=diag\{\lambda_1,\dots,\lambda_n\}$. ...
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### Solving a linear system of the form C .* (BX) = AX

I have formed the above problem, which involves a mix between element-wise (Hadamard) multiplication (indicated by ".*") and ordinary matrix multiplication. In this problem I know all of A, ...
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### Derivative of trace involving hadamard product and product of inverse matrices

I need to find the derivative with respect to $\mathbf{\Omega}$ of $$Tr\left(\left(\left(\mathbf{\Omega^{-1}}\mathbf{C}\mathbf{\Omega^{-1}}\right)\circ\mathbf{I}\right)\mathbf{S}\right)$$ In the ...
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### Has this equation $\prod_{n=1}^{\infty}(1-\frac{z}{n^a})e^{\frac{z}{n^a}+\frac{z^2}{2n^{2a}}}=a.$ solutions?

With regard to the following equation: $$\prod_{n=1}^{\infty}(1-\frac{z}{n^a})e^{\frac{z}{n^a}+\frac{z^2}{2n^{2a}}}=a,$$ I am trying to answer the following questions: for $a=\frac{\pi}{7}$, has the ...
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### Element-wise multiplication or Hadamard product

$a_{ij}$ and $b_{ij}$ elements of matrix A and B (same dimensions). I want to multiply the matrices element-wise so the resulting matrix $s$ have the same dimensions as A and B. Is this the correct ...
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### What is the Hadamard product of matrix $A$ with $BC$, for $B, C$ being matrices?

Say we have three matrices $A,B,C$. Define $\circ$ to be the Hadamard product. The usual matrix product of two matrices $B,C$ is denoted simply as $BC$. Is there an easy expression for $$A\circ(BC)$$?
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I am trying to solve a problem for my Optimization class, in which it is asked to calculate the gradient of the following function: $$g(P)=\frac{1}{2}||1_K\circ(R-Q^0P)||_F^2+\frac{\rho}{2}||Q^0||_F^2+... 0answers 27 views ### Multivariate Cauchy Integral Formula for Hadamar square of multivariate ordinary generating function. Consider an ordinary d-variate generating function of the form f(\mathbf z) = \sum\limits_{\mathbf k \in \mathbb{N}^d} a_{\mathbf k} \mathbf z^{\mathbf k}. I know that in dimension d=1, I can ... 1answer 42 views ### How to work with an expression involving matrix products and Hadamard products in least squares? With the system \boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}, where \boldsymbol{A} is a known matrix, \boldsymbol{b} is a measured vector and \boldsymbol{x} the vector we want to estimate, ... 0answers 69 views ### Mixed Product Property: Kronecker Hadamard Matrix Multiplication Given matrices A,B, C, D, I wonder whether there is a way to simplify$$(I_d\otimes A)(B\odot C)(1_d\otimes D),$$where A is n\times m, B, C are dm\times dn, and D is n\times k. Moreover,... 1answer 76 views ### Extension of the Schur product theorem to operators Given two n\times n matrices A and B, define their Hadamard product A\circ B as the element-wise product, i.e.$$(A\circ B)_{ij} = A_{ij}B_{ij}\,.$$A well known result is the Schur product ... 0answers 41 views ### Evaluating contour integrals over the unit circle of rational functions. Let p(z):= a_0 + a_1z + \cdots + a_nz^n be a degree n polynomial, let m be a large integer (which we may assume much larger than n), and let k be some integer in the range m+1, \cdots, m+n.... 0answers 103 views ### Does an integral expression exist for \xi(s)\,\xi(-s)? The product \xi^*(s)=\xi(s)\,\xi(-s), with \xi(s) the Riemann_xi_function, but ignoring the first factor \frac12, possesses some 'beauty' in the sense that it yields:$$\xi^*(s)=s^2\,(s^2-1)\,\...
Say we know that for some $b,c\in \mathbb{R}^p$, \begin{align} [(b-c)\oslash b]_j\leq \epsilon \end{align} For $A\in \mathbb{R}^{p\times p}$, is there anything we can say about \begin{align} [A(b-c)\...