Questions tagged [hadamard-product]

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

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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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Variance of the sum of two normal random variables when the variance of one is related to the variance of the other

P= $\theta$ $A^T$ X +(1-$\theta$)$A^T$ Y $\theta$ is a scalar $\sigma$ is Nx1 X and Y are both Nx1 A is Nx1 M is NxN $\rho$ is NxN $X \sim N(\mu , \Sigma _X) $ $Y \sim N(\mu , \Sigma ...
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32 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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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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16 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
52 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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24 views

Hadamard product and matrix multiplication: Algebra manipulation

I have the following equation (which comes from OLS): $C = \big((a^T \circ k^T)(a \circ k)\big)^{-1}(a^T \circ k^T)(b \circ k)$ $\circ$ is the Hadamard product and $.^T$ is the regular transpose ...
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Relate l1 norm of Hadamard product to trace

Suppose $A$ is a $q\times p$ matrix, $B$ is $q \times p$ matrix, $A_j$ is the jth column of $A$, and $B_j$ is the jth column of $B$. The following sum of $l_1$ norms, where "$\circ$" is the ...
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63 views

Derivative of a trace of a Hadamard product

Let $A$ be a $N\times5$ matrix, $\vec{b}$ be an $N \times 1$ vector and $\vec{x}$ be a $5\times1$ vector. I am looking for the derivative of the function, $$f(\vec{x}) = \text{Trace}((A\vec{x}\vec{x}^...
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Derivative of matrix product of matrix and Hadamard product of 2 matrices

Right now I'm trying to find a derivative that's stumping me: Let $A, B$ be $m \times n$ matrices and $W$ be a $p\times m$ matrix. $f = W \bullet (A \circ B)$ $\frac{\partial f}{\partial A} = ?$ (...
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Alternative formula for Hadamard product of two generating functions.

Let $f$ and $g$ be sequences of functions and $F$ and $G$ their corresponding generating functions, \begin{eqnarray*} & F(z)=\sum_{n\in\mathbb{N}} f_n z^n \\ & G(z)=\sum_{n\in\mathbb{N}} g_n ...
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152 views

Proof of Matrix Norm Inequality (Hadamard product)

Let $◦$ be the entry-wise (Hadamard) product operator, where for two matrices $$A = (a_{ij} )_{1≤i≤n,1≤j≤m}, B = (b_{ij} )_{1≤i≤n,1≤j≤m}∈ R^{n×m}$$ we define $$A ◦ B := (a_{ij} b_{ij} )_{1≤i≤n,1≤j≤m}...
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Hadamard product of two generating functions.

Let $f$ and $g$ be sequences of functions and $F$ and $G$ their corresponding generating functions, \begin{eqnarray*} & F(z)=\sum_{n\in\mathbb{N}} f_n z^n \\ & G(z)=\sum_{n\in\mathbb{N}} g_n ...
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On Hadamard product of power series and constructing a “Hadamard Algebra”

Conider two power series is definded as follows : $$ f(x) = \sum_{n=0}^\infty f_n x^n $$ $$ g(x) =\sum_{n=0}^\infty g_n x^n $$ Then , Hadamard product of $f$ and $g$: $$ (f\odot g)(x) =\sum_{n=0}^\...
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If $ B= c1^T \odot A$ is positive definite, is $A$ invertible?

Let matrix $B \in \mathbb{R}^{n \times n}$ be defined as \begin{align} B= c1^T \odot A \end{align} where $c$ is a vector with positive entries, $1^T$ is a vector of all ones, and $A$ is some square ...
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47 views

Sum of element-wise division

I want to compute the (row) sum of a matrix $A$ obtained by Hadamard division of $B$ by $C$. I found that if I were to do instead Hadamard multiplication (i.e. the standard Hadamard product, also ...
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53 views

Relationship between Hadamard and Inner Product

Suppose I have two scalars $a,b \in \mathbb{R}$ defined by the inner products: $$ a = \mathbf{a}^{T} \mathbf{t} $$ $$ b = \mathbf{b}^{T} \mathbf{t} $$ where $\mathbf{a}, \mathbf{b}, \mathbf{t} \in \...
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Solving: $\underset{\alpha}{\text{min}} \; || \left( b - A(\alpha \circ x ) \right) ||_{2}^{2}$

I want to solve the following minimization problem: $$ \underset{\alpha}{\text{min}} \; || \left( b - A(\alpha \circ x ) \right) ||_{2}^{2} $$ where $\alpha, x \in \mathbb{C}^{N}$ and $b \in \...
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relation between matrix multiplication and Hadamard product

Let $\circ$ denote the Hadamard product, i.e. entry-wise multiplication. Suppose we have 4 vectors $\vec{x}, \vec{y}, \vec{z}$ and $ \vec{t}$ be in $\mathbb{Z}^m$ such that $$ \vec{x}\circ\vec{y}=\vec{...
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1answer
120 views

Scalar-by-matrix derivative involving trace and Hadamard product

I am quite new to matrix calculus, and I am trying to find the scalar-by-matrix derivative to a seemingly simple problem but have yet to find a solution online. I am trying to find \begin{align} \frac{...
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176 views

Derivative of quadratic with Hadamard product

I proposed a similar question involving logarithms, but the problem is about scalar. I am trying to solve the more generalized form: $$ \min_{\mathbf{x} \in \mathbb{R}^N_+} \left( \sum_i \left( h_i^...
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160 views

Hadamard product and tensor

i'm pretty new at linear algebra, so here is my question: Let $T$ be a second rank tensor. Consider the next expression in basis $b$ = {$b_1, b_2, b_3$}: $B$ = $K$ $\circ$ $T$ There $K$ is a ...
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402 views

How to compute derivative with Hadamard product?

Let $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$ are $n$-dimensional column vector, and $$f = \mathbf{x}\circ \mathbf{y} \circ\mathbf{z}$$ Here $\circ$ is the element-wise Hadamard product. Then how ...
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40 views

How to solve a matrix equation involving Hadamard product (Markov Chains, Average First Passage Cost)

I am trying to find the average first passage cost of transition from $i$ to $j$ in a Markov Chain. If we consider costs of transitions as C[i,j], with the obvious property that $C[j,j]=0$, then we ...
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Expressing a matrix product of a matrix and a hadamard product as a hadamard product of matrix products

I have a matrix $\mathbf{X}\in\mathbb{C}^{N\times K}$ which undergoes a transform (specifically the ZCA whitening transform) represented by $\mathbf{W}\in\mathbb{C}^{N\times N}$ to form $\mathbf{Y} = \...
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171 views

Variance of Hadamard Product

Multivariate vectors $\textbf{X} \sim N(0, \textbf{A})$ and $\textbf{Y} \sim N(0, \textbf{B})$ Now I want to show that $Var(\textbf{X} \odot \textbf{Y}) = \textbf{A} \odot \textbf{B}$ (Matrix) (...
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1answer
87 views

Diagonal Matrices and the Hadamard Product

I am trying to show that $$ \begin{bmatrix} \Sigma{11} \\ \vdots \\ \Sigma{nn} \end{bmatrix} = (\boldsymbol{\mathbf{Q}} \odot \mathbf{Q})\boldsymbol{\lambda} $$ Where $$\boldsymbol{\Sigma} =\bf{QDQ^...
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51 views

Covariance of Hadamard product

Let I am trying to show that $$Var(\bf{X} ⊙ \bf{Y}) = Cov(\bf{X} ⊙ \bf{Y}, \bf{X} ⊙ \bf{Y}) = \bf{C}$$ where $\bf{X}$ ~ $N(0, \bf{A}) $ and $\bf{Y}$ ~ $N(0, \bf{B}) $ are two independent multivariate ...
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1answer
154 views

Proof that the determinant of a Covariance matrix is equal to the determinant of the corresponding correlation matrix times the product of variances

If $\Sigma\in \mathbb{R}^n$ is a positive-definite covariance matrix with corresponding vector of variances $v = diag(\Sigma)$ and standard deviations $s = \sqrt{v}$, then the corresponding ...
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1answer
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Invertibility of function $x \circ (Ax)$ ($\circ $ for Hadamard product)

I would like help with the conjecture that the function $f:\mathbb{R}_{\ge 0}^n\to\mathbb{R}_{\ge 0}^n $ with $f(x) = x \circ (Ax)$ where ∘ is the Hadamard product (equivalently, $f_i(x) = x_i \...
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How should my matrix (multiplication) equation look like?

Consider the following matrix: \begin{align} X=\begin{bmatrix} x_{11}\;\; x_{12} \\ x_{21}\;\; x_{22} \\ \end{bmatrix} \end{align} I want to multiply each element by $e_{ij}$, so I finally have the ...
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1answer
224 views

Differentiating scalar, product of matrix and Hadamard multiplications, applying product and chain rule?

Suppose we need to differentiate a scalar which is the product of several matrix multiplications and Hadamard (elementwise) products between matrices. $$ Y= (A(B(XC)\circ D)\circ E)F$$ $$\frac{\...
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contracting function and matrices

consider the function $$ x \in \mathbb{R}^N,\quad f(x) = x \circ \frac{Cx}{x^T C x} $$ Where $C$ is a symmetric $n \times n$ real matrix positive definite and $\circ$ is the Hadamard product. Can ...
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Decoupling Hadamard Product

Given the matrix-vector product $ A (p \circ q) $, is it possible to decouple $ p $ from the expression, such that $ A (p \circ q) = L p $?. Ultimately, I need to minimize $ || A (p \circ q) ||^2_2 ...
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63 views

Derivative of a vector difference with Hadamard power

What is the derivative of the following?: $ \frac{\partial( \hat{\mathbf{y}} - \mathbf{y})^{\circ2}}{\partial \hat{\mathbf{y}}}$ here $\mathbf{y}$ is a constant column vector of $n\times 1$ ...
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What is the physical meaning of the sum of entries of the Hadamard product of two matrices?

Suppose that I have a symmetric matrix $\textbf{A}$, which represents the strain (or deformation) matrix. Then, what is the physical meaning behind the sum of the Hadamard product of $\textbf{A}$ with ...
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1answer
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Identity involving the Hadamard product

I am trying to understand an identity involving the Hadamard product of two matrices $A_1, A_2$ over the complex numbers of dimension $n$. Define $$S = \sum_{i=0}^{N-1} (e_i) \otimes(e_i)^t \...
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Prove the Hadamard product representation

Let $A$ and $B$ be $m \times n$ matrices with low-rank structures: $$ A = U_{A}\Sigma_{A}V_{A}^{T},\quad B= U_{B}\Sigma_{B}V_{B}^{T}, $$ Prove that Hadamard product $A\circ B$ admits the following ...
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1answer
66 views

Solve $x \circ a + x \circ (Bx) + c = 0$ for $x$?

Is there a solution to $x \circ a + x \circ (Bx) + c = 0$ for $x$, where $B$ is an $N \times N$ matrix, $x$, $a$ and $c$ are $N \times 1$ column vectors, and $\circ$ is the Hadamard product (element-...
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How to simplify repeated convolution and Hadamard multiplication

I’ve determined that the following expression gives me the correct answer in a programming challenge: $$ (A \circledast M) \times H) \circledast M) \times H) \circledast M) \times H) \circledast M) \...
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1answer
79 views

Differentials to derivatives involving trace of matrices

Suppose $P$ is a real-valued function of the $p\times m$ (real) matrix $\mathbf{Q}$. After taking its differential, one arrives with the following: $$ d(P(\mathbf{Q})) = \operatorname{trace}\...
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Chain and product rule for Hadamard product differentiation

(Asked a similar question before but deleted to add further detail) Similar to this question and a related to this question, how can I apply the chain and product rule to find the Jacobian of $$ f_1(...
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1answer
65 views

Hessian of quadratic form of function using Hadamard and Frobenius notation

Related to this question, I am trying to compute the Hessian of $$ g(r, \theta) = [r\cos(\theta)]^{\top} A \, [r\cos(\theta)] = f(r, \theta) ^{\top} A \, f(r, \theta) \tag{$*$} $$ for $r, \theta \in \...
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1answer
106 views

Matrix Differentiation (involving Hadamard products)

I am trying to differentiate over the following Frobenius Norm: $$\Phi =||A-(B\circ C)D ||^2_F$$ with respect to B, C, D respectively, i.e.: $$\frac{\partial \Phi}{\partial B}, \frac{\partial \Phi}{\...
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43 views

Derivative of $tr(A(C \circ X)BB'(C' \circ X')A')$

Can we differentiate this function: $tr(A(C \circ X)BB'(C' \circ X')A')$ w.r.t $X$? Also, $tr(A(C \circ X)Y)$ w.r.t $X$.
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Determinant of Hadamard product / sum of matrices (one diagonal)

I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $...
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81 views

Hadamard product derivative

If $\circ$ represents the Hadamard product, and $^*$ the conjugate-transpose operation. Given $$f_{(\mathbf{x})} =(\mathbf{x} \circ \mathbf{x})^*H(\mathbf{x} \circ \mathbf{x}) - (\mathbf{x} \circ \...
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72 views

Hadamard product and being unitary

Let $A\in\mathbb{C}^{n\times n}$ and $A=B\circ B$ where $B$ is a unitary matrix and $\circ$ accounts for the Hadamard product. Can we say any thing about $A$ to be unitary or not?
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75 views

Diagonalization and the Hadamard product

Let $B \in \mathbb{C}^{n\times n}$ be unitarily diagonalizable such that $B=V\Lambda V^*$. Let $A=B\circ B$ where $\circ$ accounts for the Hadamard product. Then we can say that $A$ is also unitarily ...
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158 views

Duality of the convolution theorem in Fourier Domain

Convolution in the time domain can be represented as a Hadamard (pointwise) product in the Frequency domain. Using the instructions specified at https://in.mathworks.com/matlabcentral/answers/38066-...