**5**

votes

**0**answers

118 views

### Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...

**0**

votes

**1**answer

115 views

### What is the following (matrix) operator called?

Let $\mathbf{A}=\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1m} \\ A_{21} & A_{22} & \cdots & A_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ A_{m1} & A_{m2} ...

**5**

votes

**0**answers

164 views

### Non-linear positive map

In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...

**1**

vote

**0**answers

153 views

### Complexity of reordering a matrix which consists independent sub matrices

Introduction:
Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, ...

**0**

votes

**1**answer

90 views

### SO(3) transformation that produces a reflection [closed]

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
...

**2**

votes

**1**answer

136 views

### Finding matrices $A$ such that the entries of $A^n$ have specified signs

What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of ...

**2**

votes

**1**answer

163 views

### Actions of the unit circle on finite complex matrices

Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle.
How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on ...

**0**

votes

**0**answers

38 views

### Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions:
$$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$
and
$$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$
such ...

**0**

votes

**2**answers

350 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

**3**

votes

**0**answers

95 views

### Rank of a particular matrix

Denote $P_{2n}$ to be collection of homogeneous total degree $2$ real polynomials in exactly $2n$ variables such that coefficient of every monomial is either $1$ or $-1$.
Split variable set into ...

**6**

votes

**3**answers

286 views

### Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...

**0**

votes

**1**answer

150 views

### Comparison of Lp norm of matrix and its entry wise norm. [closed]

I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.

**12**

votes

**1**answer

301 views

### Expected size of determinant of $AA^T$ for random circulant and Toeplitz matrices

If $A$ is chosen uniformly at random over all possible $n$ by $n$ Toeplitz (or circulant) (0,1)-matrices, can we give any bounds for the expected size of the determinant of $AA^T$? All arithmetic is ...

**1**

vote

**1**answer

183 views

### Expected size of determinant of $AA^T$ for non-square random $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all ...

**1**

vote

**1**answer

56 views

### How to determine an unitary operator involved in an unitary transformation?

Let two real matrices $A$ and $B$ be unitarily equivalent. How to determine (computationally or theoretically) the unitary operator $U$ s.t. $A = UBU^\dagger$? Is it possible for some special class of ...

**4**

votes

**2**answers

485 views

### Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case
$A = \begin{pmatrix} ...

**3**

votes

**1**answer

200 views

### Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.
What is minimum ...

**30**

votes

**3**answers

2k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**2**

votes

**1**answer

43 views

### Information on special matrices similar to Jacobi matrices

Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...

**1**

vote

**1**answer

81 views

### Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of:
$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ...

**0**

votes

**0**answers

69 views

### Invertible matrices sending a subspace into a disjoint copy of it

Assume $\{X_i\}$ and $\{Y_i\}$ are sequences of $n\times n$ matrices defined as:
$X_0=I$, $X_1=0$ and $X_{i+1}=ABA^{-1}X_i+AX_{i-1}$ for all $i\geq1$, and
$Y_0=0$, $Y_1=I$ and $Y_{i+1}=BY_i+Y_{i-1}$ ...

**2**

votes

**1**answer

144 views

### Prove or disprove a matrix logarithm equation

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices, and let $A,B\in\Bbb{S}_{++}^n$.
Is it possible to express the logarithm of $A^{-1}B$ as a ...

**0**

votes

**0**answers

69 views

### Minimum rank of certain matrices

Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in ...

**0**

votes

**2**answers

251 views

### Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...

**3**

votes

**1**answer

137 views

### Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...

**1**

vote

**0**answers

134 views

### comparing norms of block-matrices

Let $k\in\mathbb{N}$ and let $d\in\mathcal{M}_k=\mathcal{B}(\ell_2^k)$ be a $k\times k$ diagonal matrix with strictly positive entries. Let now $x_1,\ldots,x_m\in\mathcal{M}_k\,\,(m\in\mathbb{N}$ - ...

**6**

votes

**1**answer

128 views

### Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...

**2**

votes

**1**answer

139 views

### Selecting columns of a set of boolean matrices with constraint on the ones in each row

I've come up with the following question in my research: Let $S$ be a finite set of $n \times n$ matrices with elements 0 or 1. denote $n_i$ as the total number of 1's in the $i$th row of all matrices ...

**3**

votes

**3**answers

270 views

### Finding the square root of a special matrix

Let $n\in \mathbb N$ be a natural number, $x_1,\cdots.x_n$ be formal variables. Consider the following $n\times n$-matrix $M_n:=diag\{x_1^2+\cdots+x_n^2,\cdots,x_1^2+\cdots+x_n^2\}$, can we find a ...

**2**

votes

**1**answer

171 views

### Vanishing of permanent of a Vandermonde matrix [Edited]

Does there exist an explicit criterion (or a good sufficient condition) for proving that a Vandemonde matrix:
$$A(x_1,\dots,x_n):=\left[ \begin{array}{llll}1 & x_1 &\dots& x_1^{n-1}\\ 1 ...

**2**

votes

**1**answer

107 views

### Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?

(I've asked this in MSE but nobody had an idea since dec 14...)
(Roughly related, but generalizing, of this earlier MSE question)
Background: ...

**1**

vote

**1**answer

125 views

### Row-stochasticity of the Jacobian matrix of a stationary distribution

Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...

**0**

votes

**0**answers

77 views

### Partial Vandermonde Circulant Determinant Expression

Consider following partial Vandermonde type, circulant matrix
$\begin{bmatrix}
x_1 & x_2 & 0 & \dots & 0 & x_n\\
x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\
\vdots ...

**1**

vote

**0**answers

76 views

### Perturbation of eigenvalues of some special matrices

In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...

**4**

votes

**0**answers

99 views

### Relation between Euclidean distance matrices and squared-distance matrices

Let us refer to $D$ as the matrix with $D_{ij} = ||x_i - x_j||_2$ and $H = D \circ D$ e.g. the matrix with $H_{ij} = ||x_i - x_j||_2^2$ as its entries. Are the spectra of these matrices related at ...

**0**

votes

**0**answers

47 views

### Pseudo-braided fusion categories

A few definitions first, please replace with the standard terminology (and correct me if I confuse all the by-names of fusion categories :-)
I call a complex number $z$ pseudo-cyclotomic if $|z|=1$.
I ...

**0**

votes

**0**answers

76 views

### Finding a “special” non singular submatrix

Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ ...

**3**

votes

**1**answer

275 views

### Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...

**2**

votes

**0**answers

24 views

### Is there some kind of lower bound for estimation error of the estimation of (near) low-rank matrices in high-dimension?

I'm reading S.Negahban and M.J.Wainright's paper, ESTIMATION OF (NEAR) LOW-RANK MATRICES WITH NOISE AND HIGH-DIMENSIONAL SCALING. In the paper, they give a upper bound for estimation error of the ...

**6**

votes

**2**answers

213 views

### What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $n \times n$ matrix ...

**7**

votes

**1**answer

193 views

### Determinant of some covariance matrix (Gaussian kernel process)

Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - ...

**0**

votes

**1**answer

82 views

### Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...

**6**

votes

**1**answer

268 views

### Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$.
A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...

**1**

vote

**0**answers

64 views

### Smallest sum of original column entries in 2d matrix [closed]

I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this:
...

**1**

vote

**1**answer

65 views

### Given $M$, minimize $|Mx|_0$

Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find
$\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$,
where the $\ell_0$ "norm" is measured by simply counting the number ...

**0**

votes

**0**answers

67 views

### Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...

**0**

votes

**0**answers

83 views

### Proof sought that a certain function is approximatively 3-dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$.
Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...

**1**

vote

**0**answers

171 views

### Is there a way to simplify this apparently huge characteristic polynomial calculation?

Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let ...

**8**

votes

**5**answers

630 views

### Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.

**8**

votes

**2**answers

455 views

### On closest unitary matrix

In this question $\|A\|_p$ is the normalized $p$-th Schatten norm which is defined to be $\left(\mathbb E_{i} \lambda_i^p\right)^{1/p}$, where $\lambda_i$ are singular values of matrix $A$.
Suppose ...