# Questions tagged [hadamard-product]

The hadamard-product tag has no usage guidance.

44
questions

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? ...

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, ...

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 ...

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 ...

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 ...

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{...

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_{...

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 ...

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 ...

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 $n\times n$ ...

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 ...

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}=...

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 ...

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 ...

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 ...

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, ...

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 $\odot$ ...

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 $...

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}...

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 ...

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 ...

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),...

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 ...

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-...

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 ...

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 ...

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
$Y=U\Lambda U^T$...

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 ...

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{$\...

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 ...

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 ...

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. ...

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 ...

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 ...

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}+\...

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 \\
\...

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, ...

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 ...

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 $h_{4n, k}=1$)...

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^{\...

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 $A$ ...

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, $C$ ...

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 ...

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 ...