Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level ...

learn more… | top users | synonyms (1)

1
vote
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
3answers
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
1answer
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
3answers
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
3answers
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
1answer
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
1answer
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
0answers
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
1answer
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 ...