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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1answer
943 views

Reachability in graphs using adjacent matrix

Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less ...
2
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0answers
193 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 ...
3
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1answer
488 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
1answer
195 views

adjacency matrix of random geometric graphs [closed]

Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...
7
votes
2answers
773 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 ...
4
votes
1answer
224 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 < ...
13
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2answers
2k 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 ...
6
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0answers
101 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 ...
0
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1answer
254 views

Books or references on multidimensional matrix operations [closed]

Have the 2D matrix operations been generalized to n-dimensional matrices? Are there any books that define various operations on multidimensional matrix? I'd like to see operations such as ...
5
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0answers
205 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 ...
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0answers
192 views

How to understand the matrix behind a Hamiltonian?

thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...
1
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0answers
60 views

Possible diagonal values of a product of matrices with some specific characteristics

Hello all, This is a question that might or might not be related to my previous one. Imagine you have two matrices: Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where ...
5
votes
1answer
197 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 ...
3
votes
1answer
282 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
2answers
393 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
501 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
2answers
309 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)
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0answers
87 views

Characterizing the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$, $$f(x,y) = e^{\imath\pi x g(y)}$$ where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$ ...
3
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0answers
897 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
276 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 ...
1
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0answers
154 views

Ring-theoretic version of a matrix problem

Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is ...
4
votes
2answers
263 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
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2answers
515 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 ...
17
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1answer
894 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 ...
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1answer
192 views

What is such an equation called?

Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector? $Ax+f(\lambda)Bx=g(\lambda)x$.
2
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1answer
227 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 ...
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1answer
219 views

Proof of Tracenorm Equality

Lemma 1 in this paper: http://ttic.uchicago.edu/~nati/Publications/SrebroShraibmanCOLT05.pdf claims that $\|X\|_{\Sigma} = \min_{V^TU=X} \frac{1}{2}(\|U\|_{Fro}^2 + \|V\|_{Fro}^2),$ where ...
1
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2answers
389 views

is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ?

is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ? thanks a lot!
3
votes
1answer
225 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 ...
1
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0answers
123 views

bounds on the entries of an inverse circulant matrix

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...
0
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0answers
75 views

Moore-Penrose question

Let $A=BD^{\dagger}B^{T}$. I am looking for conditions under which $A^{\dagger}$ is a "nice" expression in $B$ and $D$ and their Moore-Penrose pseudo-inverses. Do you know of such conditions?
1
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1answer
494 views

power of a block triangular matrix

I have a matrice in the form : $$M = \begin{pmatrix} A & 0 & 0 \\\ B & A & 0 \\\ C & D & A \end{pmatrix} $$ where $A,B,C,D$ are diagonalizable square matrice and I want to ...
1
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1answer
149 views

Scaling laws for singular values of random matrices

Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1. Are there any results on the asymptotic behavior of its $i$-th largest singular ...
3
votes
2answers
898 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 ...
2
votes
2answers
783 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 $\lambda_{2}-\lambda_{1}$. ...
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3answers
528 views

Number of parameters needed to specify a Hermitian matrix of rank r.

Hi, i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature. 1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank ...
0
votes
3answers
399 views

Convex Combination of 2 hermitian matrices

Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices ...
1
vote
1answer
51 views

Is the trace of a Lyaponov transform of a semistable matrix always nonpositive?

Let $A$ be a semistable real matrix (i.e. the real parts of all the eigenvalues of $A$ are nonnegative). Let $P$ be a positive definite matrix. Is it always true that $\text{trace}{A^{T}P+PA} ...
2
votes
1answer
291 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 ...
0
votes
1answer
148 views

Eigenvector localizaiton

I have raised this sort of question before but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry ...
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0answers
60 views

sharper interlacing

The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I ...
2
votes
2answers
458 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 ...
3
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2answers
422 views

S-matrix conjecture: status?

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

name for a matrix operation

If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?
3
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2answers
514 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 ...
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1answer
177 views

eigenvector update formula

Suppose that $B$ is a Hermitian matrix with one known eigenpair $(\lambda,v)$. (assume its the smallest or largest pair, if you like). Form the rank one update $B+\rho bb^{T}$. Now I'm interested in ...
2
votes
3answers
221 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 $||u-v|| \leq \kappa(A) |\frac{u^{\*}Au}{u^{\*}u}-\frac{v^{\*}Av}{v^{\*}v}|$? ...
5
votes
2answers
703 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 ...
2
votes
1answer
186 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 ...
5
votes
0answers
237 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 ...