**5**

votes

**2**answers

413 views

### Determinant of non-symmetric sum of matrices

Given three real, symmetric matrices $A\succ0$ and $B$, $C⪰ 0$.
How can it be shown that:
$$\det(A^2+AB+AC) \leq \det(A^2 +BA +AC+BC) ? \qquad (\star)$$
Where $A^2$ is symmetric and positive ...

**8**

votes

**0**answers

238 views

### Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...

**20**

votes

**4**answers

999 views

### Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...

**6**

votes

**1**answer

281 views

### Almost orthogonal vectors in subsets of euclidean space

Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of almost-orthogonal sets ...

**1**

vote

**2**answers

1k views

### Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & & ... \\\
0 & b_{2} & a & ... & 0 ...

**3**

votes

**1**answer

188 views

### Matrix norms / eigenvalues / singular values / another thing

OK, here is what is probably a stupid question.
Let $M$ be a non-symmetric real matrix: for example, the shear matrix
$\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$.
There ...

**3**

votes

**1**answer

147 views

### What is the name of this measure of matrix “degenerateness”

Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called?
If the vectors are columns in a matrix $\Phi$, then ...

**0**

votes

**0**answers

115 views

### Recognize this matrix norm?

I stumbled on the following simple matrix norm, which I haven't seen elsewhere. I wonder if it is well known, has a name, and has been studied elsewhere. The definition of this norm for a matrix A ...

**0**

votes

**0**answers

52 views

### Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries.
The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...

**4**

votes

**2**answers

291 views

### spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it?
Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...

**6**

votes

**2**answers

317 views

### Dimension of incomplete matrix over finite fields.

Hi,
Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}_{p^k}$. In fact, one knows that the diagonal entries are zero and all other entries are ...

**2**

votes

**1**answer

151 views

### On solution of a class of discrete-time Lyapunov equation

Hello members, let's consider the following equation
$$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$
where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment ...

**2**

votes

**1**answer

106 views

### On solution of a discrete-time equation

Hello, members.
I have a problem for the following problem
when I derive an optimization algorithm for stochastic singular systems
$$S(k+1)=A(k)S(k)A^{T}(k)+R(k)+F(k)S(k+1)F^{T}(k)$$
where ...

**4**

votes

**2**answers

258 views

### tracial triples

Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the ...

**2**

votes

**1**answer

161 views

### An Interesting variant of Rayleigh Quotient

Let $A$ and $B$ be two given hermitian positive semi-definite matrices, then what is the solution for
$$\max_{x\neq 0}\frac{x^HAx}{x^HBx+1}.$$
I am looking for closed form solutions.
If the ...

**4**

votes

**0**answers

134 views

### Examples of functions from matrices to real numbers with certain properties

Let $M(\mathbb{R})$ be the set of all matrices (of any size) over $\mathbb{R}$. Let $v : M(\mathbb{R}) \rightarrow \mathbb{R}$ be a function which satisfies the following 5 properties:
If ...

**1**

vote

**1**answer

86 views

### On solution of a recursion with rectangular matrices

Greetings to members here.
The question is how to calculate the solution $S(k)$ of the following recursive equation
$$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$
where $J$ and $A$ are rectangular not ...

**3**

votes

**1**answer

154 views

### Schur product, partial order

Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner partial order.
How to ...

**2**

votes

**0**answers

114 views

### Products of matrices of a certain form

Are $n \times n$ matrices of the form
$$\pmatrix{1&1&1&1 \cr x&1&1&1 \cr x&x&1&1 \cr x&x&x&1}$$
studied anywhere? I am interested in the structure of ...

**1**

vote

**2**answers

137 views

### Strictly positive definite autocovariance function of fGn

Hi,
let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter.
I want to show that $\gamma$ is ...

**0**

votes

**2**answers

87 views

### Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants
\begin{align}
w^{H}C_1w>0 \\\
w^{H}C_2w>0 \\\
...~~~~~~~~~~ \\\
...

**4**

votes

**1**answer

588 views

### Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X = (X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...

**1**

vote

**0**answers

79 views

### Matrices with a common fischer basis

Let $A$ be a real symmetric $n\times n$ matrix normalised such that $Tr[A]=1$. Define 'fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for this ...

**12**

votes

**0**answers

318 views

### Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...

**3**

votes

**0**answers

160 views

### Matrix where every subset of rows has maximal rank

I am looking for a class of matrices $M(n(m), m, k(m), \phi)$ with the following properties:
M is $n \times m$ where $n(m) > m$.
Every subset of rows of size $k$ has (maximal) rank $m$.
$n(m)$ ...

**2**

votes

**1**answer

99 views

### Increasing sequence of normal magic squares

The questions below are motivated by pure curiosity. I heard of the first question from my former advisor. I have no idea how difficult they are, since I have no experience with magic squares.
By a ...

**1**

vote

**5**answers

628 views

### Does this matrix shape have a name?

I'm using a lot of matrices that look like this:
$$A_3 =
\begin{bmatrix}
a & b & b\\
b & a & b\\
b & b & a
\end{bmatrix}
$$
i.e. the diagonal entries are all the same, and all ...

**3**

votes

**4**answers

865 views

### Diagonalizing a Complex Symmetric Matrix

For a symmetric matrix M with complex entries, I want to diagonalize it using a matrix A, such that
$AMA^T = D$, where D is a diagonal matrix with real-positive entries.
Question 1: When can this ...

**0**

votes

**1**answer

444 views

### How to compute difference between 2 similarity matrices?

Hello,
I have two n*n correlation matrices with values ranging between -1,1. (2 correlation matrix because I have the same n terms under 2 different conditions)
I then transformed the correlation into ...

**2**

votes

**1**answer

528 views

### Inversion of complex matrix

Hello all,
Assume matrix of complex numbers described as a sum of real matrices $A$ which is diagonal and $B$ which is symmetric (and block symmetric if the term is correct):
$A+ Bi$
I want to ...

**5**

votes

**1**answer

231 views

### The minimal norm of a shifted stochastic matrix

Hello,
Given a row-stochastic matrix $M$ with singular values $\sigma_{1}\geq\ldots\geq\sigma_{n}$, I am looking for an upper bound on the expression: $\min_{\alpha}\parallel M- ...

**9**

votes

**0**answers

441 views

### Is “being a full ring of quotients” a Morita invariant property?

Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...

**1**

vote

**1**answer

148 views

### integral basis of orthogonal complement

Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$.
My goal is to find an ...

**2**

votes

**0**answers

192 views

### eigenvalues of the sum of a stochastic matrix and a diagonal matrix

Let $D$ be a real diagonal matrix $D=diag(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real, row-stochastic ...

**2**

votes

**1**answer

478 views

### Non symmetric matrices with real eigenvalues

Consider the following block matrix
$A=\pmatrix{A_1 & A_2\cr kA_2^\top & A_3}$
where $A_1$ is a symmetric matrix, $A_3$ is diagonal matrix and all entries of $A$ are real and non-negative.
...

**3**

votes

**2**answers

338 views

### Square submatrix

We have $2n\times 2n$ binary matrix with $k$ of its elements are $1$. We are searching for an $n\times n$ submatrix full of $1$s.
What is the least $k$ such that we can always find one?
What is ...

**3**

votes

**1**answer

373 views

### Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...

**12**

votes

**6**answers

520 views

### Invertibility of a certain matrix indexed by the Hamming cube

For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim
On submeasures on Boolean algebras, arXiv 1212.6822v3
and in Section 7 the ...

**2**

votes

**1**answer

155 views

### How to find the nilpotent submatrices of a symmetric, real matrix?

Given a symmetric, real $n \times n$-matrix $M$, is there a way to find all $m \times m$-submatrices ($1 < m < n$) that are nilpotent?
By the Cauchy interlacing theorem, I know that $M$ must ...

**9**

votes

**1**answer

348 views

### Probability that a random distance function is metric

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...

**2**

votes

**1**answer

395 views

### Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries

I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...

**2**

votes

**2**answers

175 views

### Primitive orthogonal vectors/Unimodular matrices

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice.
A set of integer vectors ...

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

**2**

votes

**1**answer

184 views

### Multiple eigenvalues over imperfect fields

Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\alpha(A)$ for the ...

**0**

votes

**1**answer

186 views

### Covering the cone of positive semidefinite matrices by intervals

Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices?
How about a general convex cone?
For the finite case the ...

**2**

votes

**4**answers

368 views

### Nth root of a matrix as an analytic function?

Let $A$ be a $k \times k$ invertible matrix over complex numbers.
If it possible to write its nth root as an analytic function (i.e. power series in $A$)?
EDIT: Complex coefficients can be functions ...

**5**

votes

**0**answers

159 views

### Largest entry of the inverse matrix?

I wonder if there is a "qualitative way" of predicting from the structure ix of the matrix $A$ which entry of $A^{-1}$ will be the largest. I am specially interested in the case
that $A$ is a ...

**0**

votes

**0**answers

148 views

### Summation of eigenvalues of tri-diagonal matrix smaller than specific value

Is there any analytic expression for summation of eigen-values of a tri-diagonal matrix which are smaller than a constant value? Or even a rough approximation for it. How about case of a general ...

**1**

vote

**1**answer

362 views

### Sum of elements of inverse matrix

Hello all,
Assume NxN matrix A of complex values. I want to calculate the sum of all elements of its inverse.
The problem is that calculating the inverse is computationally expensive and since I am ...

**0**

votes

**2**answers

257 views

### Is it possible to obtain the vectors orthogonal to a given one by orthogonal transformations?

Hello, everyone!
Supposing that there is a unit vector in $n$-dimensional real space $\mathbf{x}_1\in\mathbb{R}^n$, I want to get a group of $n-1$ vectors to form an orthogonal basis with ...