# Questions tagged [hadamard-product]

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37
questions

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56 views

### An elementary proof of Davies' inequality

In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality.
Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...

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84 views

### Full-rank Hadamard product given a certain structure

Let us assume that we have a full-rank randomly chosen $k\times (m\cdot l)$ matrix, $\boldsymbol{H}$, with
$l \leq k \leq (m\cdot l)$ and no specific structure (e.g., a realization of an IID complex ...

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54 views

### Eigenvector of Hadamard matrix functions

Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...

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38 views

### Hadamard-like product on infinitely differentiable functions

Has the following operation $*$ on formal power series $f,g$ been studied before?
$$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$
where $n$ is a nonnegative integer? This is the typical Hadamard ...

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172 views

### Riesz equivalent of Riemann hypothesis and Hadamard product

First define Hadamard product :
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 $g$:...

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**1**answer

105 views

### Properties of eigenvalues and eigenvectors of a particular random matrix

Let $\mathbf{A}$ be a given $n \times m$ matrix with positive entries, and $\mathbf{B}_{n\times m}$ be a random i.i.d complex Gaussian matrix with unit variance. Assume that $\mathbf{C}$ is the ...

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224 views

### Sum of inverses of zeros of $L$-functions

Let $L(s)=\sum_{n\ge1}a(n)/n^s$ be a "standard" $L$-function, say converging
for $\Re(s)>k$, having an analytic continuation with no poles to an entire
function of order $1$ with functional ...

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**1**answer

400 views

### Solving $AXB + X\odot C = D$

I need to solve the following equation for $X$ with $d$-by-$d$ matrices $A,B,C,D$ and Hadamard product $\odot$
$$AXB + X\odot C = D$$
Vectorizing all terms gives a solution with $O(d^6)$ complexity, ...

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50 views

### Upper bound on the maximal eigenvalue of a matrix given the eigenvalues of Hadamard power of it

I have a matrix $A$ which has -1,1 outside the diagonal, and 0 on the diagonal. One can assume 1 on the diagonal(because all the eigenvalues can be related).
I am looking for an upper bound on the ...

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118 views

### Matrix completion problem with determinant condition?

Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and an integer $m$ is there a polynomial in $n$ algorithm to find a $\{-1,0,+1\}$ matrix $T$ such that $$\mathsf{det}(L+T\odot M)=m$$ where $\odot$ ...

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333 views

### Zeros of the Hadamard product of holomorphic functions

Let $A(z) = \sum_{n=0}^{+\infty}a_n z^n$ and $B(z) = \sum_{n=0}^{+\infty}b_n z^n$ be two formal power series with complex coefficients. The Hadamard product of $A$ and $B$ is the formal power series $...

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103 views

### Preserving the strictly total positivity of special bases by using radial basis functions

Let $\left[\alpha,\beta\right]$ be a non-empty interval and consider an $\left(n+1\right)$-dimensional subspace $\mathbb{S}_{n}$ of $C^n\left(\left[\alpha,\beta\right]\right)$ such that $\mathbb{S}_{n}...

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122 views

### Convergence of sequence of images of Schur multipliers

Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}_{\geq 0})=\ell^{2}$ (by bounded operator I will always ...

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149 views

### References for the theory of Hadamard functions and compositions of random vectors

Recently, I fell in love with the pointwise/elementwise/componentwise/Hadamard/Schur functions and compositions of random vectors such as Hadamard squares and products of random vectors. Here is one ...

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87 views

### Non singularity of a generalised Vandermonde matrix through Hadamard product

I'm currently trying to prove the following.
Consider $k_1,\dots,k_N$ complex numbers not lying on the real and imaginary axes. Then consider
\begin{equation}
W_N(x)= \text{Wronskian}\big(\cosh(k_1x),...

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**1**answer

271 views

### On the notion of Hadamard rank of matrix

Given a matrix $M\in\Bbb F_2^{n\times n}$, define its Hadamard rank $h(M)$ to be the minimum number of rank $\leq2$ matrices in $\Bbb F_2^{n\times n}$ with Hadamard product (that is, the entry-wise ...

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109 views

### On the complexity of writing down matrices

Consider families of $0/1$ matrices in $\Bbb B$ where $1+1=1$:
$\mathcal M_{1,n,c}$ contains $2^n\times 2^n$ matrices that can be written as Hadamard product of $t=O(2^{(\log n)^c})$ matrices $$(J_n-...

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258 views

### Rank of Hadamard product with random matrices

I do research in statistics and am not sure whether the following is considered research level or not in mathematics. If it isn't, I'm happy because that means the answer is probably known and I can ...

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103 views

### Behavior of matrix rank under thresholding of its elements

Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is ...

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243 views

### Eigendecomposition of the Hadamard product of a rank one symmetric matrix and a positive definite symmetric matrix

Is it possible to say anything about the eigenvalues and eigenvectors of a matrix
$X = Y \circ xx^T$
where $Y$ is a positive definite symmetric matrix with known eigen-decomposition
$Y=U\Lambda U^T$...

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**4**answers

584 views

### The rank of a perturbed triangular matrix

$\DeclareMathOperator{\rk}{rk}$
The question below is implicit in this MO post, but I believe it deserves to be asked explicitly, particularly now that I have some more numerical evidence.
Suppose ...

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690 views

### Pointwise (Hadamard) matrix product and the rank

$\DeclareMathOperator{\rk}{rk}$
Suppose that $A$ is a square matrix of order $n$. If, for any polynomials $P$ and $Q$ with $\deg P+\deg Q\le 2$, we have
$$ P(A)\circ Q(A^t) = P(1)Q(1)\, I_n \tag{$\...

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**1**answer

467 views

### Maximum eigenvalue of Hadamard power of a positive semidefinite matrix

Let $K$ be a covariance matrix. It is positive semidefinite, its diagonal elements are all 1, and its off-diagonals are between -1 and 1. Let $K.^2$ be its element-wise power (Hadamard power). Can we ...

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170 views

### infinitely many non-trivial zeros for $L(s,\chi)$ using Hadamard product for $\xi(s,\chi)$

Here is the definition of $\xi(s,\chi)$:
$\xi(s,\chi)= \left(\frac{s(s-1)}{2} \right)^{1_{\chi=1}} (q/\pi)^{\frac{s+a}{2}} \Gamma \left( \frac{s+a}{2} \right) L(s,\chi)$
Here is the definition of ...

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495 views

### Integral of the entrywise square of the exponential of a matrix

Note: I posted my question on math.stackexchange but got no answer. That is why I am asking it here.
Let $A$ be a $n\times n$ square matrix such that the real part of all eigenvalues are negative. ...

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91 views

### sub-space restricted minimum eigenvalue of Hadamard product of two PSD matrices

Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{n\times n}$ be two positive semidefinite matrices. Also let $\mathbf{A}\circ \mathbf{B}$ denote the Hadamard product of $\mathbf{A}$ and $\mathbf{B}$. A ...

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190 views

### Matrix equation and Hadamard product

I have a matrix equation involving square real matrices of the form:
$$X(X^{T} \odot A) + (X \odot A)X^{T} = B$$
where $\odot$ denotes the Hadamard product. How can I determine a solution for X?
Any ...

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961 views

### Matrix equation with Hadamard product and its own inverse involved

I know there is an almost exactly same question here but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\...

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**1**answer

273 views

### eigenvalues of a generalization of Hadamard product matrix

I have a Laplacian matrix ($L$), which is positive semi-definit. Then I have this matrix
$$\Delta=\begin{bmatrix} \delta_{11} & \ldots & \delta_{1n} \\
\vdots & \ddots & \vdots \\
\...

**3**

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**1**answer

96 views

### Billera Tree Space

I am studying the tree space of Billera and I do not really understand why it is an Hadamard Space. I have already read L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, ...

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1k views

### Hadamard Product and Eigendecomposition

I just found this related question in here Q1.
Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...

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129 views

### Prime Hadamard Matrices

Assume that $n$ is a sufficiently large number. Is there a Hadamaed matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$)...

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**1**answer

454 views

### A Hadamard product of the zeros of the Riemann integral. Does it put any constraints on where the $\rho$'s can reside in the critical strip?

I have deleted a previous, now obsolete question on the same topic.
Take the well-known Riemann integral:
$$\displaystyle \pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\...

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**1**answer

704 views

### How to solve a matrix equation with both inverses and a hadamard product?

I have a matrix equation of the form:
$$
A^{-1} = B + A \circ C
$$
where $\circ$ denotes the Hadamard product (i.e., $(A\circ C)_{ij} = A_{ij}B_{ij}$). How can I determine if a solution for $A$ ...

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**1**answer

502 views

### Inequalities for Hadamard products of complex symmetric matrices

Consider a complex symmetric matrix $$ C= C_R + i C_I $$ with $C_R,C_I \in \text{Mat}_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C_R$ are all strictly positive. Then, $C$ ...

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352 views

### Hadamard product and inertia

One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...

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864 views

### Generalizations of Oppenheim's inequality

The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...