Questions tagged [hadamard-product]

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

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54 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 ...
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2answers
33 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 ...
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1answer
122 views

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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higher order hadamard product

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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52 views

Recursive chain rule

Given the following equations: $$ \begin{aligned} o_t&=\sigma(x_t, h_{t-1};W_o) \\ \tilde{c}_t&=\tanh(x_t, h_{t-1};W_g) \\ f_t&=\sigma(x_t, h_{t-1};W_f) \\ i_t&=\sigma(x_t, h_{t-1};W_i)...
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1answer
22 views

Matrix Derivative of F-norm with Hadamard Product

I'm trying to solve $\nabla_X \| A \odot(B-X^\top C) \|_F^2$, but I don't know how to solve this... Could anyone help? Thank you in advance for any help you can provide.
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1answer
40 views

Derivative for Masked Matrix Hadamard Multiplication

In deep learning, such an operation is common: $$A = B\circ (C>0.2)$$ where $A,B,C\in \mathbb{R}^{n\times m}$, $\circ$ denotes Hadamard Multiplication and $C>0.2$ is the matrix where each ...
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1answer
46 views

Matrix times Vector where the elements are vectors

Whats the correct operation to calculate the "product" of matrix $A$ of the size $M \times L$ $$A= \begin{bmatrix} \vec{A}_{1,1} & \vec{A}_{1,2} \\ \vec{A}_{2,1} & \vec{A}_{2,2} \\ ...
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1answer
28 views

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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2answers
67 views

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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17 views

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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1answer
54 views

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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36 views

Product of a Matrix and its Rescaled Inverse

I have the following question. Suppose you have a matrix $$\mathbf{B}=\begin{bmatrix} 1 & \beta_{12} & \beta_{13} & \cdots & \beta_{1n} \\ \beta_{21} & 1 & \beta_{23} & \...
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1answer
53 views

Hadamard product of two matrices as a matrix multiplication

I have encountered the following problem: I have two $N$-by-$N$ complex valued matrices $A, B$, and then I form a third matrix as a Hamadard product of the previous two: $C = A \odot B$, so that for ...
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1answer
21 views

Expression for combining 3d and 2d matrices into resulting 2d matrix

I would like to know if there is a convenient mathematical expression for the following. I want to construct a (n x n) matrix $\mathbf{C}$ for which i know that each column $\mathbf{C}_i$ is equal to $...
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31 views

Hadamard Product inside a norm

Is it true to say $\|A\circ B\| \le \|A\| \|B\|$? $\circ$ is the Hadamard product.
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33 views

Differentiating an expression involving Hadamard Product

I'm trying to calculate $\frac{dY}{dW}$ where: $$Y = (W^TX)oC$$ 'o' denotes the hadamard product. W, X and C are matrices with dimension (d, c), (d, n) and (c, n) respectively. From matrix cookbook:,...
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1answer
30 views

Can you rearrange this simple Hadamard & Matrix product?

I have the following matrix operation: ${((AB^T) \odot C)D}$ where juxtaposition denotes the matrix product and $\odot$ the Hadamard product. A, B & D are all matrices of shape (3, 2) and C is a ...
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1answer
43 views

A question about the derivative of trace involving Hadamard product

Assume that $X\in\mathbb{R}^{n\times n}\geq 0$, $Y\in\mathbb{R}^{n\times k}\geq 0$ and $Z\in\mathbb{R}^{k\times n}\geq 0$. Let us define the function $f(Y,Z)$ as follows: $$f(Y,Z)=\Vert X-YZ\Vert_W^2:=...
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1answer
56 views

Derivative of trace involving Hadamard product

Let us assume that $A, S\in\mathbb{R}^{n\times n}$, $U\in\mathbb{R}^{n\times k}$, and $V\in\mathbb{R}^{n\times k}$. I am trying to differentiate the following expression: $$\Phi(U,V)=\mathrm{trace}\...
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1answer
73 views

Find the gradient (with respect to a matrix) of an expression containing a Frobenius norm and a Hadamard product.

I'm struggling with taking the gradient with respect to (w.r.t) the matrices $H_R$ and $H_I$ in the following expression $$\left\| Z - I\odot(H_R^TA + H_I^TB) \right\|_F^2 + \left\|W-\begin{pmatrix} ...
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1answer
33 views

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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1answer
28 views

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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1answer
33 views

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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1answer
24 views

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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1answer
55 views

Gradient of an optimization function - Frobenius norm and Hadamard product

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+...
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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 ...
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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, ...
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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,...
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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 ...
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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 $m+1, \cdots, m+n$....
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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)\,\...
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40 views

Hadamard Division: Same Linear Transformation in Numerator and Denominator

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)\...
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64 views

Hadamard product of a generating function and a sequence

I have a series $\{g_n\}$ whose values are hard to compute, but I calculated a generating function for it (I know the square root is unconventional, but it results in a nice exponential function): $$ \...
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1answer
133 views

Matrix differential of a trace with Hadamard product

I'm encountering difficulties taking the differential of the following matrix expression with respect to $S$: $\text{logdet}(S) + \text{Tr}[C(D\odot((AS^{-1/2}B)(AS^{-1/2}B)^{T}))]$ $C$ and $D$ are ...
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1answer
49 views

Given orthogonal matrix $P$, is $P \circ P$ invertible?

Let $P$ be a orthogonal matrix, i.e., $P^T P = P P^T =I.$ Then can we say that $P \circ P$ is invertible? P.S: $A \circ B$ is the elementwise product of matrices $A$ and $B$.
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1answer
67 views

Is there a general formula for $x^{T}Ax \cdot y^{T}Ay$?

Let $x,y \in \mathbb{R}^{n}$. Let A be a $n \times n$ positive-definite symmetric matrix. Is there a general formula for $x^{T}Ax \cdot y^{T}Ay$? For example, let $x = \begin{bmatrix} 2 \\ 2 \end{...
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1answer
55 views

Is $(I \circ A - I \circ B)$ positive semi-definite if $A$, $B$ and $A - B$ are positive semi-definite?

Let $A$ and $B$ are positive definite and positive semi-definite matrices, respectively. $A - B$ is positive semi-definite. Is it true that $(I \circ A - I \circ B)$ is positive-semidefinite? I ...
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2answers
83 views

Taking the matrix derivative of the product of one matrix and a Hadamard Product.

Consider the three matrices $\mathbf{C}$, $\mathbf{A}$, and $\mathbf{T}$. The matrix $\mathbf{C}$ has $\mathit{m} \times \mathit{k}$ entries, $\mathbf{A}$ is a $\mathit{k} \times \mathit{n}$ matrix, ...
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1answer
91 views

Gradient of matrix operations

Assume we have a matrix $M\in \mathbb R^{t\times qt}$, a vector $p \in \mathbb R^{r}$, and a vector $z \in \mathbb R^{qt}$, . Note that $\otimes$ is a Kronecker product and $\odot$ is a Hadamard ...
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1answer
58 views

Solving Matrix Equations including nested Hadamard products

How can we solve the following equation for B? $$yl^T \odot D = [(AB \odot D)ll^T]\odot D $$ $y_{n*1}$, $l_{d*1}$ are vectors, $A_{n*k}$, $B_{k*d}$, $D_{n*d}$ are full rank non-square matrices.
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1answer
27 views

Differentiating a scalar, product of matrices, with respect matrix

Let $\zeta$ be an $n*1$ vector: $$\zeta=[(GXB)\odot D]l$$ where $G$ is an $n* n$ matrix, $X$ is an $n* k$ matrix, $B$ is a $k* d$ matrix, $D$ an $n* d$ matrix and $l$ a $d*1$ vector We need to ...
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1answer
63 views

Minimize $x^TAx$ with respect to $T$, where $A=T\odot T^{-1}$

\begin{array}{ll} \text{minimize} & x^TAx\\\quad T\in\mathbb{R}^{n\times n}\\ \text{subject to} & A=T\odot T^{-1}\\&T>0\end{array} where the symbol $\odot$ denotes the elementwise/...
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1answer
69 views

Stuck on matrix derivative

I am stuck with this (probably simple) derivatives: $$ \frac{\partial}{\partial X}Tr((A\odot(B^{T}XB))C)\;\;and \;\;\frac{\partial}{\partial X}Tr((A\odot(B^{T}XX^{T}B))C) $$ where $A,B,C$ are ...
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1answer
95 views

derivative of hadamard product of

I am having difficulties to compute the derivative of the following expression: $$ (xx^T)\circ A $$ with respect to $x \in R^K $ where $A \in R^{K\times K}$. Although Derivative of Hadamard ...
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1answer
96 views

Derivative of L1 norm of Hadamard product

I am trying to find the derivative of $f(B)=\lambda\Vert W \bigodot B \Vert_1 + \frac{\rho}{2}\Vert A-B \Vert_F^2 + tr(\Delta^T(A-B))$ with respect to B. where B is (n×n)matrix, W is (n×n)constant ...
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1answer
312 views

Stuck on equation with Hadamard product and transpose

I have been stuck the last few days on the resolution of an equation I built for a phenomenon's modelling. Let $D_O$ and $D_H$ be square, diagonal, $n \times n$-sized matrices. Let also $H$, $O$, $W$ ...
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1answer
18 views

Matrix from scaled vectors notation

I'm trying to find a concise way to notate the following operation: $$ \vec{a}^T \space \text{(Operator)} \space \vec{b} = \begin{bmatrix} a_1 \vec{b} \space \dots \space a_N \vec{b} \end{bmatrix} $...
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1answer
74 views

Real Hadamard powers of matrices

Let $A$ be an entrywise nonnegative matrix and for any $r>0$, $A^{\circ r} = [a_{ij}^{r}]$. My question is what properties of matrix $A$ are preserved for all $r>0$ or for all $r$ in some ...

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