The matrix-theory tag has no wiki summary.

**16**

votes

**1**answer

566 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

**2**answers

834 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 ...

**11**

votes

**2**answers

818 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 ...

**7**

votes

**1**answer

767 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 $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...

**6**

votes

**2**answers

436 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

**1**answer

208 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

**0**answers

94 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

**2**answers

683 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

**1**answer

159 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

**0**answers

188 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

**2**answers

243 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

**1**answer

185 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

**2**answers

434 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

**0**answers

191 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 ...

**4**

votes

**0**answers

460 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 ...

**3**

votes

**2**answers

462 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

**2**answers

316 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

**2**answers

233 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

**1**answer

146 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

**2**answers

620 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

**1**answer

217 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

**2**answers

404 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

**1**answer

169 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) ...

**2**

votes

**2**answers

231 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

**1**answer

202 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}
...

**2**

votes

**2**answers

344 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

**1**answer

162 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

**1**answer

215 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

**3**answers

207 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

**2**answers

312 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

**1**answer

323 views

### S-matrix conjecture: status?

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

**2**

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**0**answers

151 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

**1**answer

218 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

**1**answer

176 views

### name for a matrix operation

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

**1**

vote

**3**answers

329 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 ...

**1**

vote

**4**answers

249 views

### inverse-closed matrix spaces

Is there a known characterization of such spaces?
An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the ...

**1**

vote

**1**answer

187 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?
$Ax+f(\lambda)Bx=g(\lambda)x$.

**1**

vote

**2**answers

197 views

### small sums of entries in submatrices - strange phenomenon

Suppose that $x \in \mathbb{R}^{n}$ is a vector of small positive fractions, i.e. $x_{i} \approx \frac{1}{n}$. The exact values are unknown. I form the matrix $M=diag(x)-\frac{xx^{T}}{2}$ which is a ...

**1**

vote

**1**answer

197 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**

vote

**2**answers

343 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!

**1**

vote

**1**answer

48 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} ...

**1**

vote

**1**answer

265 views

### Cholesky decomposition of a positive semi-deﬁnite

We know that a positive deﬁnite can be done for Cholesky decomposition,but I want to know that how a positive semi-deﬁnite be done for Cholesky decomposition?The following sentences come from a ...

**1**

vote

**1**answer

366 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 ...

**1**

vote

**1**answer

333 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**

vote

**1**answer

138 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 ...

**1**

vote

**0**answers

251 views

### how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
...

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193 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 ...

**1**

vote

**0**answers

108 views

### Books or references on multidimensional matrix operations

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 ...

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vote

**0**answers

45 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 ...

**1**

vote

**0**answers

68 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 $0\leq x < \infty$
...