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Questions tagged [hadamard-product]

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

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Matrix square root of squared correlation matrix

Setup: Given $y \sim N(0,\Sigma)$, suppose we want to transform $y$ to a new space so entries have zero covariance. We can use the inverse square root and apply to transform $\tilde{y} = \Sigma^{-1/...
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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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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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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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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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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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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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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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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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Conditionally (almost) definite matrices and Hadamard product

There seem to be different uses of the terminology. One says that a matrix $M$ is almost definite if $x'Mx=0 \Rightarrow Mx=0$. But here I am referring to a different one, that is there exist some $A$ ...
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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-...
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Derivation Numerical Method with partial derivatives, vectors, matrices and scalar product

I need help in finding a way to combine the equations \begin{equation} \frac{\partial J}{\partial W} \cdot \delta W = \langle Y_M^T (\eta^{'}(Y_M W) \odot (\eta(Y_MW)-C)),\delta W \rangle \end{...
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Derivative of Hadamard product with respect to matrix

I'm trying to calculate this derivative wrt matrix $F_{i}$ and simplify the whole expression: $ \frac{ \partial J_{term 1}}{\partial \mathbf{F_{i}}}= 2 (\sum_{j:G(i,j)=1} (\mathbf{W}_{i,j} \...
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Hadamard product and minimality of operator norm

Let $Y \in \mathbf{R}^{n \times n}$ be a given matrix, and let $1 \leq r \leq n$ be an integer. Suppose that $A \in \mathbf{R}^{n \times n}$ is such that $$ \|X - Y\| \geq \|A - Y\|, \quad \text{...
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Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$?

Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$? Here $\circ$ is the Hardamard product and $\|•\|_2$ is the Frobenius norm.
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Conditions for Hadamard to distribute with matrix multiplication

Let $A$ be an $n \times m$ matrix, and let $\circ$ be the Hadamard product. What are sufficient conditions on $A$ for the following to be true for all $m$-vectors $x$ and $y$? $$ Ax \, \circ \, Ay = A(...
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Solution to $Mx-x^2=0$ where $x^2$ is the square of the elements of vector $x$

I have been trying to find the solutions for $$Mx=x\circ x$$ where $\circ$ is the element wise product. One solution is $x=0$. But there is another solution $x\neq 0$, if $M$ has a real positive ...
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Frobenius Norm of Hadamard Product and Trace

I'm trying to relate the Frobenius Norm of a Hadamard Product to a trace that does not include another Hadamard Product, if possible. In other words, if A and B are (sxr) matrices, with not all ...
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Solve System of Quadratic Equations with Hadamard Product from pde collocation

I have the following system of equations: $ A x - z \cdot B x \cdot B y + c = 0$ $D y - z \cdot B x \cdot B y + d = 0$ Where I want to solve for $ x,y \in \mathbb{R}^N$. $A$, $B$ and $D$ are all ...
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matrix gradient of the Hadamard product

What is the matrix gradient for the function $||A(B \circ X) ||_F^2 $ with respect to $X$. Here $A,B \in \mathbb{C}^{n \times n}$ and $X \in \mathbb{R}^{n \times n}$.($\circ$) is the Hadamard ...
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Prove that $\|\mathbf{A}\circ uv'\|_F = 1$ when $\mathbf{A}$ is sign matrix

I want to prove that $$ \|\mathbf{A}\circ uv'\|_F = \|u\|_2 \, \|v\|_2 = 1 $$ when $\mathbf{A}$ is sign matrix. My proof is as follows ($:$ denote the Frobenius product $$ (\mathbf{A}\circ uv'):(\...
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rank inequality for Hadamard product

How I can show that $$ \operatorname{rank}(A\circ B) \leq \operatorname{rank}(A)\operatorname{rank}(B) $$ where $A,B$ are rectangular matrices and $\circ$ is the Hadamard product between the two. ...
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Derivative of a Matrix Expression with Hadamard

I have the following expression at hand: $1_{nx1}^T (X_{nxk} \circ X_{nxk})(Y_{kx1} \circ Y_{kx1})$ $X$ is a matix, $Y$ is a vector, and $1$ is a vector of ones with indicated dimensions. $\circ$ is ...
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Can Hadamard (schur) product $A \circ B$ be positive definite, if $A \succeq 0$ (positive semi-definite) and $B \succ 0$ (positive definite)?

Dear Linear Algebra experts, According to Schur Product Theorem, if both $A \succ 0$ and $B \succ 0$ are positive definite, then the Hadamard product of $(A \circ B) \succ 0$ is also positive ...
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Matrix notation for element-wise raising to the power of $n$

The Hadamard product $A \odot B$ gives the element-wise multiplication of matricies $A$ and $B$. How do I denote the raising a matrix to the power $n$, element-wise?
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Derivative of trace involving inverse and Hadamard product

Let $A, B$ be symmetric $(n \times n)$ matrices and let $A$ be invertible. I am looking for the derivative $$ \frac{\partial}{\partial A} \operatorname{tr}[A^{-1}(A \odot B)], $$ where $\odot$ is the ...
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232 views

Derivative of Frobenius norm of Hadamard Product

I am trying to find: $\frac{\partial}{\partial A} \left||Q\circ C \right||^2_F $ $\quad$ and $\quad$ $\frac{\partial}{\partial B} \left||Q\circ C \right||^2_F $ where $C= B A^T$ C is (pxq) ...
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Determinant defined as Product of Columns

Let $N$ be a non-singular matrix, $v_i$ be the column $i$ of $N$, and $M$ be a matrix with $e_i$ as columns. $M$ and $N$ have the same dimensions. I do not understand how $|\det(N)|=\prod_i ||v_i|...
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Check Reasoning On Calculation Involving Diagonal Matrix and Matrix and Hadamard Products

I apologize in advance, as I kind of realize this is a dumb question. But I need a little more mathematical rigor to my naive logic. Suppose I have an $n \times n$ diagonal matrix, $\mathbf{A}$, ...
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201 views

solve matrix equation involving Hadamard products

I'm trying to solve the equation $$ (\Sigma \circ C)^{-1} \circ C = \left[ (\Sigma \circ C)^{-1} S (\Sigma \circ C)^{-1} \right] \circ C $$ for $\Sigma$ or $\Sigma \circ C$, where $\circ$ denotes the ...
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Relation for the determinant of a special Hadamard product.

Inspired by this very related question. Let $A$ be a circulant matrix and $X$ an anti-circulant matrix of size $n\times n$. Moreover let the sum of each row in $X$ be zero, $\sum_i x_{ij}=0$. It ...
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inequality on matrix Hadamard Products $\|A \odot X\|_F$

I have two matrix with same size $A$ and $X$, $A$ is a binary matrix. $X$ is nonnegative matrix. Is there any inequality show that $\|A \odot X\|_F <= f(A) \|X\|_F$, how to find $f(A)$? my goal is ...
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Bounding the determinant of principal sub-matrices of the Kroneker product

I have a matrix $A$ that is 2 dimensional and has negative determinant. I have a matrix $B$ that is 2 dimensional and has a positive determinant. Both have strictly positive elements. I want to show ...
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366 views

Hadamard product of a positive semidefinite matrix with a negative definite matrix

If I have a positive semidefinite matrix $A$ and a negative definite matrix $B$, is it true that their Hadamard product $A\circ B$ is negative semidefinite? Ideally I am looking for a proof / a ...
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what is transposition Hadamard product?

$$F=||Q \circ X||_F^2 $$ where $\circ $ is hadamard product. How I can convert it to style of general matrix multiplication?
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When does Hadamard product rank upper bound hold?

We know $rank(A\circ B)\leq rank(A)rank(B)$. Under what conditions do we have equality?
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How to deal with hadamard product of matrix with vector

$$Ax=b$$ $A$ is a real square symmetric matrix with all non-zero entries, $x$ and $b$ are vectors. The hadamard inverse of $A$ is defined $$A^{\circ-1} \circ A = E$$ where $E$ is a matrix of ones....
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Differentiating trace of matrix product when matrix elements are functions of a vector

According to a well known formula (Eqs. 100-104 here) $$\frac{\partial}{\partial B} tr(AB)=A^T$$ For square real-valued matrices $A,B$. For simplicity assume these matrices are symmetric. But... 1)...
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139 views

Derivative of trace of Hadamard and dot product

Im struggling solving an equation and I have tried to find solution in the matrix cookbook but did not find a clue. How can I calculate the derivative of the equation which is a combination of ...
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1answer
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How to solve the linear equation $A\circ (XB) + CX = D$

How to solve the follow equation $X$ is the variable: $A\circ (XB) + CX = D$ where $ A \circ B$ is element-wise product or Hadamard product. if the $A = 1_{n \times n}$. the above equation become $(...
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1answer
114 views

Second order gradient of a non-linear element-wise function and a Hadamard product

I am applying the mean value theorem to an analysis, but I am running into problems computing the second derivative. let: $$ J = d^Tf(Wx) $$ with $f$ a generic continuous and differentiable element-...
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Is sum of inverse of max a kernel?

For $X$ be a nonempty set, a function $f :X\times X\to\mathbb R$ is called a kernel on $X$ if for all $m\in \mathbb N$ and all $x_1,\cdots,x_m \in X$ $$K_f\equiv\begin{bmatrix} f(x_1,x_1) & \cdots ...
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287 views

Algorithm for computing Hadamard product of two rational generating functions

If I have two generating functions $A(x) = \sum_na_nx^n$ and $B(x) = \sum_n b_nx^n$ then the Hadamard product is $(A \star B)(x) = \sum_{n} a_nb_nx^n$. Now when $A$ and $B$ are both rational ...
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261 views

Derivative of Hadamard product with functions

I'm new to matrix derivatives, and I'm having a bit of trouble with this one in particular. I have this equation for the function: $f(x) = M(g(x) ∘ g(x))$ Where M is a non-square matrix, '$∘$' is ...
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71 views

A is positive definite, B is positive semidefinite and all the diagonal element is positive, prove that Schur product their is positive definite

$A$ is positive definite, $B$ is positive semidefinite and all the diagonal element is positive, prove that $A\circ B$ is positive definite It is easy to prove it with Oppenheim Inequality: ...
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1answer
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Differentiate hadamard product of square matrix $S \odot VV^T \in R^{n \times n} $ over rectangular matrix $V \in R^{n \times r}$

I want to differentiate $f = \log\det(L)$ over $V$ where $L = S \odot VV^T$. The thing that I know is $df = L^{-T} : dL = (S \odot VV^T) : (dS \odot VV^T+ S \odot (VdV^T+dVV^T ))$ where $:$is a ...
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1answer
195 views

Distributing matrix-vector product across a element-wise (hadamard) product

Given this equation, how do you solve for $c_3$ (in terms of $M$, $c_1$ and $c_2$)? $(Mc_1 + e_1) \odot (Mc_2 + e_2) = Mc_3 + e_3$ I'm not sure how to algebraically distribute across the hadamard ...
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213 views

Can anyone tell me what is the derivative of hadamard product

I want to know the derivative of ${\bf{(}}{{\bf{w}}^{\mathop{\rm H}\nolimits} }{\bf{A}} \odot {\bf{(}}{{\bf{w}}^{\mathop{\rm H}\nolimits} }{\bf{A)^{\bf{*}} - }}{{\bf{g}}^H}{\bf{)(}}{{\bf{A}}^H}{\bf{w}}...