Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to [tag:linear-algebra]). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur ...

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17
votes
1answer
837 views

2x2 subdeterminants of a matrix

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B. Given then all these 2x2 determinants of an ...
13
votes
2answers
1k views

Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello, I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it. I can ask it in two different ways. Perhaps depending on the reader, the ...
12
votes
2answers
1k views

Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ...
8
votes
2answers
328 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 ...
8
votes
1answer
146 views

A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem. Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
8
votes
1answer
275 views

Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
7
votes
1answer
787 views

A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity: Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
7
votes
0answers
79 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
6
votes
1answer
318 views

On the positivity of matrices

For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called positive, if $x^{T}M x\geq 0$ holds for all non-negative real $x_1,x_2,\cdots,x_n$, where $x=(x_1,x_2,\cdots,x_n)^T$. ...
6
votes
2answers
704 views

Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere. Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...
6
votes
1answer
223 views

A generalization of van der Waerden's conjecture

I am wondering if the following generalization of van der Waerden's conjecture is true. Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T_i$. ...
6
votes
0answers
100 views

Rational points with small denominator in $U(n)$

Fix integers $n,d>0$. (I'm probably thinking about $n\leq 6$ and $d\leq 2000$.) Let $X$ be the set of matrices $A\in U(n)$ such that the entries of $dA$ lie in $\mathbb{Z}[i]$. Is there an ...
5
votes
2answers
702 views

A question about matrices with more details

Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that ...
5
votes
2answers
241 views

Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
5
votes
1answer
184 views

Matrix Inverse with Same Principal Minors

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$, and index set $\langle n\rangle = \{ 1, \dots, n \}$, and the submatrix $A(\alpha)$ with the columns and rows of $A$ with indices $\alpha ...
5
votes
1answer
126 views

Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
5
votes
0answers
202 views

On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix. Suppose there exists ...
5
votes
0answers
224 views

concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
4
votes
2answers
257 views

system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$. One of my friend asked me the following ...
4
votes
1answer
214 views

Lower bounds on matrix eigenvalues

Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that $$ 0 < \alpha < \mathrm{Re}(\mu_1) < \dots < ...
4
votes
2answers
491 views

Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $GL_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More precisely, the ...
4
votes
1answer
102 views

Calculating the dimension of the algebra generated by some given matrices

Let $X_1, X_2, \ldots, X_d$ be $n \times n$ matrices over some field $K.$ I want to calculate the dimension of the unital algebra generated by $X_1, X_2, \ldots, X_d$ for some examples in a problem I ...
4
votes
1answer
74 views

What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
3
votes
2answers
503 views

matrices whose entries sum to zero

Let $A$ be a non-singular matrix and let $s(A)$ be the sum of its entries. Under which conditions can it be assured that $s(A) \neq 0$? if you like, you can assume that $A$ is symmetric. Here is an ...
3
votes
2answers
381 views

Is Ryser's conjecture on permanent minimizers still open?

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$. Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...
3
votes
2answers
278 views

On a determinant inequality of positive definite matrices

Assume that $B$ and $A$ are two positive definite matrices. Take $B^*$ a block diagonal matrix with block $B_{11}$ and $B_{22}$ of $B$. This means the following: $$ B=\left[\begin{array}{ll} ...
3
votes
2answers
298 views

A short question about the DFT matrix

Is the DFT matrix the unique* unitary matrix with all entries of same magnitude? (*up to some trivial transformations)
3
votes
2answers
414 views

S-matrix conjecture: status?

Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
3
votes
1answer
213 views

When are cones of matrices “generated” by vectors?

The cone $P_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over ...
3
votes
2answers
854 views

Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.

Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( http://en.wikipedia.org/wiki/Wishart_distribution ). I have a ...
3
votes
1answer
446 views

What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it): ...
3
votes
2answers
196 views

Stabilization of the pencil of skew symmetric matrices by the orthogonal group

Good morning everybody. During my researches I've come across the following question. Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the ...
3
votes
2answers
488 views

Average size of determinants of integer matrices?

I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the entries of $M$ are ...
3
votes
1answer
263 views

A spectral radius inequality

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) ...
3
votes
0answers
73 views

When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form \begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} ...
3
votes
0answers
819 views

Matrix Transpose Similarility

A famous problem in linear algebra is that "A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$." I know one proof using Smith Normal form. However, I want to know an ...
2
votes
2answers
272 views

On matrix norms

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way: $|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$. Suppose we define a different function of ...
2
votes
2answers
203 views

quadratic matrix equation

Find all symmetric matrices $X=X^{T}$ such that \begin{align} XDX^{T}=-D \quad (1) \end{align} where $D\ne 0$ is a real diagonal matrix. For example, $X=iI$ satisfies $(1)$. Can you get a ...
2
votes
2answers
433 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 ...
2
votes
1answer
180 views

Simultaneous decomposition of three projectors

A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal ...
2
votes
1answer
276 views

Is there a natural distance between skew hermitian matrices?

Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to ...
2
votes
3answers
219 views

'Condition number' for Rayleigh-Ritz quotient

Suppose that $A$ is a Hermitian matrix and that $u,v$ are two vectors. Is there some known function $\kappa(A)$ so that $||u-v|| \leq \kappa(A) |\frac{u^{\*}Au}{u^{\*}u}-\frac{v^{\*}Av}{v^{\*}v}|$? ...
2
votes
2answers
678 views

Estimating a spectral gap

Suppose you have a real positive definite matrix $A$ who eigenvalues are $\lambda_{1} \leq \lambda_{2} \leq \ldots \leq \lambda_{n}$. I am interested in bounding from below $\lambda_{2}-\lambda_{1}$. ...
2
votes
1answer
79 views

Distribution of the $\alpha$-parameter of a $2\times 2$ Haar-distributed, unitary matrix

It is well known that any $2\times 2$ unitary matrix $\mathbf{U}$ can be parametrized as $$\mathbf{U}=\begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{j}\beta_1}\end{pmatrix} \begin{pmatrix} ...
2
votes
0answers
153 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
2
votes
0answers
82 views

Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent \begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation} of a 3-tensor $W_{j,k,l}$ of ...
2
votes
0answers
505 views

eigenvalues of a symmetric tridiagonal matrix with zero diagonals

I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements): $$ \left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a ...
2
votes
0answers
182 views

Inverse of sparse matrix is not generally sparse [closed]

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the ...
2
votes
1answer
224 views

Asymptotic Behavior of Non-Analytic Function of the Eigenvalues

Hello, Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$. If $A_n$ were a sequence of Hermitian ...
1
vote
1answer
180 views

name for a matrix operation

If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?