Questions tagged [hadamard-product]
For questions about Hadamard product between two matrices, or it can concern analytic functions.
245
questions
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0
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9
views
How do you calculate the degree of a polynomial involving Hadamard products?
If I have the following polynomial with an indeterminate vector $\vec{x}$, how do I compute the degree of let's say $\vec{x} \cdot \vec{x}$? (with $\cdot$ being the Hadamard product)
- 185
4
votes
1
answer
78
views
Series of product of legendre polynomials with shifted degree
I am working on some quantum mechanics and I would love to find a closed expression for the series
$$
S(x,y) = \sum_{l=0}^\infty P_l(x) P_{l+1}(y)
$$
$x,y \in \left[ -1, 1\right]$ and also for its ...
-1
votes
0
answers
19
views
Is interlacing of zeroes of parabolas preserved by Hadamard power?
Knowing that two parabolas in general form:
$$f(x)=x^2+ax+b \qquad g(x)=x^2+cx+d$$
have real,interlacing zeroes (i.e. $x_1<y_1<x_2<y_2$), is it true that $\forall p>1, p\in \mathbb{R}$
$$f(...
- 11
1
vote
1
answer
81
views
Solve matrix equations involving vectorization and Kronecker product
I want to find solutions for matrices $A\in \mathbb{R}^{m\times n}$ and $B\in \mathbb{R}^{n\times m}$ in the following equations:
$$
\left\{
\begin{matrix}
A^TM_1 = A^T(AB\odot M_2)
\\
...
- 13
0
votes
0
answers
8
views
Generalized eigenvalue problem with real symmetric matrices and Hadamard product
I have a generalized eigenvalue problem of the form
$$
(S \circ A) v = \lambda S v
$$
where $\circ$ denotes the elementwise or Hadamard product and $S$ and $A$ are real symmetric matrices
As far as I ...
- 163
0
votes
1
answer
65
views
Matrix equation involving a Hadamard product?
I'm trying to find out the solution to B for the following matrix equation:
$$
A^TM_1=A^T(AB\odot M_2)
$$
where $A\in \mathbb{R}^{m\times n}$, $B\in \mathbb{R}^{n\times m}$, and $n<m$.
$M_1, M_2\in ...
- 13
0
votes
0
answers
14
views
Derivative of determinant of matrix product of matrix and Hadamard product of 2 matrices
My problem is how to compute the partial derivative of $
f(
\boldsymbol{\Lambda}_{\mathsf{h}},
\boldsymbol{\Lambda}_{\mathsf{x}},
\boldsymbol{\Lambda}_{\mathsf{z}}
)
$ with respect to $
\boldsymbol{...
- 1
1
vote
1
answer
57
views
Correctly accounting for Hadamard product
I have an equation of the following form:
$$ \frac{\partial a}{\partial t} = Da + Fe^{i\mu \theta}$$
where $a = a(\theta,t)$ is a function and $D$ is a linear operator.
When discretized for solving ...
- 403
1
vote
1
answer
42
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How to prove Hadamard product rank upper bound hold?
In enter link description here,
@Ben Grossmann mentioned that if one of the matrices has rank 1 and no non-zero entries, $rank(A\circ B)= rank(A)rank(B)$.
My progress is as follows, where the ...
- 11
1
vote
0
answers
109
views
Eigenvectors of Hadamard product of a positive definite matrix and a symmetric positive semidefinite matrix
Let $b \in (0,1]$ and $n<\infty$. Consider the following square matrices:
Symmetric positive semi-definite matrix $A \in M_{n \times n}(\mathbb{R})$, and
$B \in M_{n \times n}(\mathbb{R})$ with $(...
- 11
4
votes
0
answers
73
views
A probabilistic proof of Oppenheim's inequality?
Oppenheim's inequality is a standard result about the Hadamard product of positive definite matrices. It goes as follows, let $A=(a_{ij})_{i,j\leq n},B=(b_{ij})_{i,j\leq n} \in S_n^{++}$ where $S_n^{++...
- 687
-1
votes
1
answer
48
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Trace inequality of Hadmard product
For any Hermitian matrix $\bf A$ and invertible matrix $\bf B$, how do I derive the inequality below
$${\rm tr}(({\bf B}^H{\bf A}{\bf B}) \circ({\bf B}^H{\bf A}{\bf B}))\geq \lambda_{\min}^4({\bf B}){\...
- 21
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votes
0
answers
108
views
Hadamard product of two sets of vectors?
For any two vectors $x,y$ in $\mathbb{R}^N$, the Hadamard product (element-wise multiplication) is defined by
$$x\circ y\equiv(x_1y_1,\cdots,x_Ny_N)\in\mathbb{R}^N.$$
I am curious if anyone has ...
- 785
1
vote
1
answer
97
views
How to solve the problem of trace optimization which includes Hadamard product?
I have the following minimization problem, where I want to find W,
\begin{align}
&\min \mathrm{tr} (((W^TK)\circ(W^TK))^T((W^TK)\circ(W^TK))L)\\
&\text{s.t.} ~ W^TKHKW = I
\end{align}
where $\...
- 11
0
votes
0
answers
26
views
Determinant of Hadamard power of an all-positive matrix
Suppose $C$ is $n\times n$ positive definite matrix with $C_{ii}=1$ and off-diagonals between zero and one $0<C_{ij}<1$. what can be said of determinant of $C^{\odot k}$ in this special case? In ...
- 639
0
votes
1
answer
51
views
How to generate a 3D matrix from a 2D matrix through column-wise Hadamard product?
I have a matrix defined as,
$
{\bf G}=[{\bf g}_1 \quad {\bf g}_2\quad...\quad{\bf g}_N],
$
where ${\bf g}_i$ is a column vector of the length $N$. The tensor is defined as,
${\bf M}(i,j)={\bf g}_i \...
- 13
1
vote
0
answers
35
views
What can one say about the eigen spectrum of the Hadamard or pointwise product of two matrices A and B
Can we give bounds or infer any type of information about the spectrum of the point-wise product of two matrices $A$ and $B$ given knowledge about the spectrum of $A$ and $B$ ?
Are there non trivial ...
- 167
0
votes
1
answer
79
views
$n$th-order Hadamard power (or root) of a vector
I am familiar with the notation of the $n$th-order Hadamard power
$$
\mathbf{A}^{\circ n}
$$
or root
$$
\mathbf{A}^{\circ \frac{1}{n}}
$$
I wonder if it is sensible to use this notation for vectors as ...
- 285
0
votes
1
answer
24
views
Matricial Equation with both Pointwise and Standard Multiplication
I am having trouble simplifying an equation that contains both pointwise (Hadamard) and standard matricial multiplication.
Given that $I_{n\times1}$ and $K_{n\times1}$ are real $n\times1$ matrices; $...
- 1
0
votes
0
answers
112
views
Does matrix multiplication distribute over the Hadamard product?
I know that the Hadamard product is distributive over addition. But suppose that $A, B \in \mathbb{R}^{n \times n}$ and $v \in \mathbb{R}^n$. Then can we say
$$v^T (A \circ B) v = v^TAv \cdot v^TBv$$
...
- 359
0
votes
1
answer
113
views
Hadamard (element-wise multiplication) product rank
I am having some problems on understanding an inequality regarding the rank of the Hadamard product (element-wise product). I have $B=A\circ A$ where $A$ is a $n\times r$ matrix, and $\circ$ is the ...
- 5
1
vote
2
answers
63
views
Solve system for elements of a matrix
I have a system of $n$ equations which follows a particular pattern as follows (showing the case $n=3$):
$$\phi = a_1 + \psi_2 a_2 + \psi_3 a_3 \\
\phi = \psi_1 a_1 + a_2 + \psi_3 a_3\\
\phi = \psi_1 ...
- 137
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votes
0
answers
27
views
Interpolation from unevenly distributed points of function with compactly supported Fourier transform
It is not so hard to prove the Nyquist-Shannon theorem. It basically says a real valued function $f$ having a compactly supported Fourier transform function $\hat f$ can be interpolated precisely from ...
- 9,438
2
votes
0
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22
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Is there a generalization of this formula for the determinant of the Hadmard product
Let $A$ and $B$ be any complex $2 \times 2$ matrices. Then a short calculation gives that
$$ \det(A \circ B) = \frac{1}{2}\left( \det(A) \operatorname{perm}(B) + \operatorname{perm}(A) \det(B) \right),...
- 5,134
11
votes
1
answer
357
views
An invertible matrix is orthogonal if and only if the inverse is equal to the transpose on nonzero elements
Let $A$ be an invertible real matrix, and suppose that $(A^{-1})_{i,j} = (A^{T})_{i,j}$ whenever $(A^{T})_{i,j}\ne 0$. Is it true that $A$ is orthogonal?
I found this statement in a paper without ...
- 10.4k
-2
votes
2
answers
131
views
What is the gradient of $x \mapsto \frac{1}{2}(x^2)^\top A(x^2)$?
Let matrix $A$ be sparse and symmetric but not semidefinite. Since I would like to use projected gradient descent, I must find the gradient of $x \mapsto \frac{1}{2}(x^2)^\top A(x^2)$, where $x^2 = \...
0
votes
1
answer
56
views
Derivative of Hadamard Product Multiply by Summing Vector (each element is 1)
Given $\mathbf{A} \in \mathbb{R}^{k \times n}$, diagonal matrix $\mathbf{W} \in \mathbb{R}^{k \times k}$,
$$
\mathbf{F}(\mathbf{X}) = \mathbf{W}(\mathbf{AX} \odot \mathbf{AX}) \mathbf{1}_{3}
$$
where $...
- 35
1
vote
0
answers
40
views
System of second-degree polynomial equations ("Hadamard vector polynomial equation")
I am interested in solutions of the following system of polynomial equations that can be written as an 'element-wise' vector polynomial:
$$
\mathbf{0} = \mathbf{A}^{(0)}\mathbf{1} + \mathbf{A}^{(1)}\...
- 19
0
votes
0
answers
15
views
What set of matrices $X,Y$ satisfy $X \circ Y = X P Y$ for at least one permutation matrix $P$?
Suppose we have three $n \times n$ matrices $X,Y,P$ where $X,Y \in \mathbb{R^{n \times n}}$ and $P$ is a permutation matrix. Let us take $\circ$ to be the element-wise product between two matrices.
...
- 1,746
0
votes
0
answers
67
views
"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 ...
- 61
2
votes
3
answers
95
views
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:
$$\...
- 23
0
votes
1
answer
42
views
Computation of two Jacobians
Definitions
Consider the following function $f:\mathbb{R}^N\mapsto\mathbb{R^2}$
\begin{equation*}f(\ell_{k+1})=
\left[\begin{array}{c}
v_{k+1}\,\cos(h_{k+1})\\ v_{k+1}\,\sin(h_{k+1}) \end{array}\...
- 481
2
votes
1
answer
324
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Inverse of matrix with hadamard product
Let $A$ and $B$, $X$ be matrices with $\mathbb{R}^{n \times n}$ where $A$, $B$ are a dense and sparse matrix, i.e., the almost elements of $B$ are zeros, respectively. I'm looking for a way to solve ...
- 21
3
votes
1
answer
381
views
Lower bound on smallest eigenvalue of hadamard product of two Hermitian matrices
Let A and B be $n \times n$ real symmetric matrix. Suppose A is positive definite and denote its smallest eigenvalue as $\lambda_{\min}(A)>0$. All elements of B are positive and bounded, i.e. $0\le ...
- 113
0
votes
2
answers
174
views
Matrix operation to exponentiate each element in a vector
I am using the following matrix algebra to obtain a vector, however, I eventually need all the resulting elements to be exponentiated.
\begin{equation}
\begin{split}
\boldsymbol{\beta}^{\textsf{T}}\...
user963084
0
votes
0
answers
87
views
Conditional convergence of the Hadamard product
The product
$$\prod_\rho \left(1-\frac{s}{\rho}\right)$$
where $\rho$ ranges over all zeros of the Riemann zeta function is often referred to as the Hadamard product (because its convergence was first ...
- 360
1
vote
1
answer
164
views
Upper-bound for nuclear norm of $A \circ (v \otimes v)$ in terms of operator norm (or nuclear norm) of matrix $A$ and $L_\infty$-norm of vector $v$.
Let $A \in \mathbb R^{n \times }$ be a psd matrix such that $\|A\|_{op} \le r_1$ and $\|A\|_{*} \le r_2$. Let $v \in \mathbb R^n$ such that $\|v\|_\infty \le r_3$. Let $B:=A \circ V$ be the Hadamard ...
- 9,140
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votes
1
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294
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Chain rule and derivative with matrix product?
I'm trying to compute some derivatives with given vectors and functions: column vector $X=[x_1,x_2,\dots,x_n ]^T$ and $Z=[z_1,z_2,\dots,z_n ]^T$, row vector $Y=[y_1,y_2,\dots,y_n ]$
$f(X,Y)=e^{XY}$
$$...
- 11
2
votes
1
answer
174
views
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 ...
- 319
1
vote
1
answer
58
views
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 $\...
- 101
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votes
0
answers
48
views
(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})...
1
vote
1
answer
152
views
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 ...
- 387
5
votes
1
answer
1k
views
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 $\sum A \odot B$ ...
- 151
0
votes
1
answer
202
views
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 ...
- 11
3
votes
0
answers
136
views
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/...
1
vote
0
answers
35
views
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 ...
- 321
0
votes
1
answer
127
views
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 ...
- 575
0
votes
1
answer
85
views
$\sum(F\circ F)\geq x_1^2+\cdots+x_n^2$
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
$\sum(F\circ F)\geq x_1^2+\cdots+x_n^2$, ...
- 2,947
2
votes
2
answers
91
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 ...
- 1,746
1
vote
2
answers
179
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 ...
- 21