**1**

vote

**1**answer

237 views

### Upper bound for sum of absolute values of eigenvalues of Hermitian matrix

Given a hermitian, but not necessarily positive, sparse matrix $C = (c_{ij}) \in \mathbb{C}^{n \times n}$ and $n \ggg 1$ ($n \approx 2^{100}$) with eigenvalues $\lambda_1 \le \lambda_2 \le \dots \le ...

**2**

votes

**0**answers

124 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**15**

votes

**2**answers

625 views

### Matrix equation $XAXBXC=I$

Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution ...

**2**

votes

**1**answer

194 views

### Trace of a Product of Finitely Many Matrices with Cosine Entry

Can someone help me prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
2\cos\frac{2j\pi}{n} & -m \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & ...

**2**

votes

**0**answers

41 views

### Generate matrices at the given distance from the initial matrix

I have n-by-n correlation matrix $C_{ij}$, it's symmetric positive definite, and $C_{ii}=1$ and $C_{ij}\le1$ etc.
I want to generate a bunch of matrices $M_{ij}$ which are on the same distance from ...

**0**

votes

**0**answers

63 views

### $l_{\infty}$ norms of matrix perturbations

Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that ...

**2**

votes

**1**answer

157 views

### Algebraically independent matrix invariants

Let $V$ be the space of pairs of $n \times n$ matrices over $\mathbb{C}$ and let $G$ be the space of $n \times n$ permutation matrices which acts on $(A,B) \in V$ by simultaneous conjugation. It is ...

**5**

votes

**0**answers

144 views

### Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$

Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is
\begin{equation}
\Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in ...

**4**

votes

**2**answers

91 views

### Integral roots of circulant matrix

When does the circulant matrix have only integral roots?
For example: all roots of the adjacency matrix of the complete graph $K_n$ are integer, which its adjacency matrix is circulant, but in case ...

**0**

votes

**2**answers

129 views

### The condition number of a scaled Vandermonde matrix

Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...

**0**

votes

**0**answers

50 views

### Sparse Matrix Reordering

Matrix reorderings are important for many direct solvers. Sometimes the objective
is to reduce the bandwith or the generated fill in by LU Decomposition.
I am interested in a reordering which reduces ...

**0**

votes

**1**answer

125 views

### Reference request: Strong Connectivity and the Incidence Matrix

Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix?
Details: Consider a digraph $(V, E)$ with vertex set
$$V = ...

**3**

votes

**0**answers

284 views

### Maximizing sum of matrices

Over the last few months, I've been trying to find the solution to a research-related problem I'm having. However, my research is not in mathematics, and my progress toward reaching a solution has ...

**1**

vote

**1**answer

186 views

### Limit of largest eigenvalue [closed]

For positive definite matrix, if we increase the dimension to the infinity, is it true that the largest eigenvalue stays bounded from above?
In other words does the following limit exists:
...

**0**

votes

**1**answer

242 views

### Meaning of $[A,B]$ when $A$, $B$ are self-adjoint

This is just a question about notation, but it got no useful answers on math.stackexchange.
Let $L$ be the Lie algebra of $n\times n$ Hermitian matrices, with Lie bracket $(A,B)\mapsto i(AB-BA)$.
...

**6**

votes

**1**answer

162 views

### Varieties parametrizing skew-symmetric matrices

Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices.
Let $M\in\mathbb{P}(\bigwedge^2V)$, for any ...

**9**

votes

**3**answers

492 views

### Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...

**8**

votes

**1**answer

770 views

### Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...

**12**

votes

**3**answers

505 views

### The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their eigenvalues, what about pairs?

For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of ...

**0**

votes

**1**answer

93 views

### How to write a given rank matrix with some constraints?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.
...

**3**

votes

**1**answer

176 views

### Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$?
My intention ...

**2**

votes

**2**answers

132 views

### What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?

**3**

votes

**1**answer

186 views

### Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$

I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...

**2**

votes

**1**answer

180 views

### Complexity of sparse matrix-vector multiplication?

I have a vector $\mathbf{x}$ of size $m\cdot n$ of zeros and ones, i.e., $\mathbf{x}\in\{0,1\}^{m\cdot n}$ and a matrix $\mathbf{A}$ of size $\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)$ of ...

**8**

votes

**2**answers

908 views

### Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...

**1**

vote

**0**answers

376 views

### Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...

**1**

vote

**0**answers

48 views

### Most general hypotheses to have matrix-exponential as solution of linear matrix-ODE? [closed]

What are the most general hypotheses under which the exponential function $\exp(-B(t))$, with $B(t)$ the indefinite integral of a real Lebesgue-integrable $d\times d$-matrix function $A(t)$, solves ...

**2**

votes

**1**answer

108 views

### Enumerator Polynomials for Linear Anytime Codes

Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...

**5**

votes

**0**answers

154 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

122 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

173 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

158 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

94 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

166 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

47 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

353 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

98 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

300 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

176 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

310 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

186 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

66 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

780 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

207 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

3k 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

47 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

95 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

70 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

149 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 ...