Questions tagged [hadamard-product]

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

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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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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: $$\...
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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}\...
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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 ...
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Lower bound on smallest eigenvalue of hadamard product of two Hermitian matrix

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 ...
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Maximum of the permanent of the Hadamard product between a unitary matrix and its permuted matrix

The problem is to $$\max_{U}\left|\operatorname{perm}\left(U\circ U_{\sigma}\right)\right|$$ where $U=(u_{i,j})$ is an $n\times n$ unitary matrix and $U_{\sigma}=(u_{i,\sigma(j)})$ is its permuted ...
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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}}\...
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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 ...
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How to find the gradient of matrix multiplying hadamard product [closed]

I'm trying to find the gradient of A(x∘x) with respect to x, where ∘ is the Hadamard product and A is a matrix with positive real values. Thanks in advance!
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Hadamard product of two non-zero vectors with infinite and zero elements

Did anybody ever define (generalize) the Hadamard product of two non-zero vectors $\mathbf{a}\circ \mathbf{b}=\mathbf{c}$ in such a way that if $a_n=0$ and $b_n=\infty$, then $c_n=e^{-\gamma}$? For ...
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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 ...
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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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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 ...
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Convex sets of matrices

I am looking for any information about matrix sets $S$ that verify the following convexity property: $A, B \in S, 0 \leq \lambda \leq 1 \implies [A(i,j)^{1-\lambda} B(i,j)^\lambda]_{ij} \in S$. Do ...
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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 $\...
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Bounding coefficients of $c + xf(x) = (x-r)(f \bigodot g)(x)$ (Hadamard product)?

Suppose we have two power series $f(x), g(x)$ with integer coefficients, say all but a finite amount of them positive. and an expression of the form: $c + xf(x) = (x-r)(f \bigodot g)(x)$ where $\...
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(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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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 ...
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1 answer
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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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How to express $\sqrt{A^TA}$ in terms of $A$

I have a Gramm matrix $G=A^TA$. Is there any simple way to express $\sqrt{G}$ in terms of $A$ using standard matrix operations, like product, transposition, inverse, element-wise function $f(A)$, ...
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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 $\sum(F\circ F)\geq x_1^2+\cdots+x_n^2$, ...
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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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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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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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1 answer
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Matrix Derivative of F-norm with Hadamard Product [closed]

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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1 answer
102 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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1 vote
1 answer
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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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1 answer
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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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2 answers
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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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2 votes
1 answer
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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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1 answer
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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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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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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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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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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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2 votes
1 answer
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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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2 votes
1 answer
93 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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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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1 answer
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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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1 answer
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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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3 votes
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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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1 vote
1 answer
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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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1 vote
1 answer
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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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