Questions tagged [hadamard-product]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
35 views

Lower bound of the smallest singular value of Hadamard power

Suppose that $A \in \mathbb{R}^{m \times n}$, does the lower bound of the smallest singular value of hadamard power of $A$ ($A^{\circ 2} = A \circ A, A^{\circ 3} = A \circ A \circ A, \cdots$) exist? ...
user avatar
  • 21
7 votes
2 answers
507 views

If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?

Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class. Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
user avatar
2 votes
2 answers
222 views

When is $(I - X)^{-\top} \circ X = 0$?

I am currently looking at the following expression $(I - X)^{-\top} \circ X = 0$, where $\circ$ is the Hadamard product, $\top$ is the transpose, $I$ is the identity, and $X$ is non-negative and ...
user avatar
  • 59
2 votes
1 answer
155 views

Singular value of Hadamard product

Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0$, $Var(A_{i j}) = 1/n$ for any $i,j$. $B$ is an $n \times n$ symmetric matrix with $B_{ii} = 0$. I need to find a upper bound of ...
user avatar
  • 21
2 votes
1 answer
216 views

The rank of the Hadamard product

For matrices $D\in C^{d×p}$ and $E\in C^{d×p}$ with $d>p$, if $D$ is a full column matrix, for what condition that $D \odot E$ is also a full column matrix where $\odot$ denotes the Hadamard ...
user avatar
1 vote
1 answer
350 views

Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers

When counting the number of integers $n(x)$ below a certain non-integer number $x$, the following series could be used: $$n(x) = x-\frac12 + \sum_{n=1}^{\infty} \left(\frac{e^{x \mu_n}} {\mu_n}+\frac{...
user avatar
  • 3,979
3 votes
1 answer
131 views

A Hadamard product representation for Keiper's $\tau$-function?

In this paper J.B. Keiper defined the following function: $$\tau_k = \sum_{j=1}^k (-1)^j\,{k-1 \choose j-1} \sigma_{j+1} \qquad k \ge 1, k \in \mathbb{N} \tag{1}$$ where $\displaystyle\sigma_r = \sum_{...
user avatar
  • 3,979
1 vote
0 answers
156 views

About the Hadamard conjecture

On the wikipedia article about Hadamard Matrix it says that "The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is $92$" But it also says that ...
user avatar
20 votes
1 answer
748 views

Hadamard factorization of L-functions

I have already asked this question here in a different form, but really need an answer. Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc... (Selberg ...
user avatar
  • 8,928
2 votes
0 answers
107 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 $n\times n$ ...
user avatar
1 vote
0 answers
102 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 ...
user avatar
  • 49
2 votes
0 answers
167 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}=...
user avatar
  • 21
1 vote
0 answers
46 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 ...
user avatar
2 votes
1 answer
161 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 ...
user avatar
  • 129
2 votes
0 answers
268 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 ...
user avatar
  • 8,928
10 votes
1 answer
508 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, ...
user avatar
2 votes
1 answer
130 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 $\odot$ ...
user avatar
  • 12.8k
3 votes
3 answers
482 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 $...
user avatar
1 vote
0 answers
123 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}...
user avatar
  • 31
2 votes
1 answer
129 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 ...
user avatar
  • 31
1 vote
0 answers
169 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 ...
user avatar
2 votes
0 answers
117 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),...
user avatar
2 votes
1 answer
322 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 ...
user avatar
  • 12.8k
1 vote
0 answers
113 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-...
user avatar
  • 12.8k
3 votes
0 answers
357 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 ...
user avatar
  • 131
2 votes
2 answers
114 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 ...
user avatar
  • 2,016
2 votes
0 answers
356 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 $Y=U\Lambda U^T$...
user avatar
19 votes
4 answers
662 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 ...
user avatar
  • 21.3k
12 votes
0 answers
828 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{$\...
user avatar
  • 21.3k
1 vote
1 answer
550 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 ...
user avatar
1 vote
1 answer
195 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 ...
user avatar
5 votes
3 answers
745 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. ...
user avatar
  • 542
0 votes
1 answer
121 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 ...
user avatar
  • 859
2 votes
0 answers
213 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 ...
user avatar
  • 21
3 votes
2 answers
1k 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}+\...
user avatar
2 votes
1 answer
343 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 \\ \...
user avatar
  • 29
3 votes
1 answer
107 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, ...
user avatar
1 vote
1 answer
2k 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 ...
user avatar
  • 322
0 votes
0 answers
153 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 $h_{4n, k}=1$)...
user avatar
4 votes
1 answer
637 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^{\...
user avatar
  • 3,979
3 votes
1 answer
868 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 $A$ ...
user avatar
8 votes
1 answer
547 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, $C$ ...
user avatar
  • 123
3 votes
2 answers
394 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 ...
user avatar
5 votes
2 answers
1k 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 ...
user avatar