Questions tagged [nonnegative-matrices]
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23
questions with no upvoted or accepted answers
10
votes
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answers
401
views
A conjecture of Blakley and Dixon about odd powers of positive matrices
In a 1966 paper Blakley and Dixon conjecture the following. Let $S$ be a symmetric matrix with nonnegative entries and let $u$ be a unit vector with nonnegative entries. For integers $k\ge j$ both odd,...
- 505
8
votes
0
answers
394
views
Eigenvalues of cyclic stochastic matrices
Let's consider the following $n \times n$ cyclic stochastic matrix
$$ M= \begin{pmatrix}
0 & a_2 & & & &b_n \\\
b_1 & 0& a_3& &&& \\\
& b_2 & ...
- 181
6
votes
0
answers
99
views
What is this matrix decomposition called and does it exist always? - II
Given a rank $2r$ matrix $M\in\Bbb Q_{\geq0}^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank at most $r$ such that $M=M_+-...
- 13.7k
5
votes
0
answers
426
views
Non-diagonalizable positive matrices
Let $n\geq 3$ and $E_n$ be the set of $n\times n$ matrices $A$ satisfying the $3$ following properties:
$\bullet$ its entries $(a_{i,j})$ are positive integers.
$\bullet$ the eigenvalues of $A$ are ...
- 3,574
5
votes
0
answers
226
views
An inequality concerning non-negative integer matrices with constant row and column sums
[I posted this question on math.stackexchange a few weeks back, but no luck there so far: https://math.stackexchange.com/questions/1095659/an-inequality-concerning-non-negative-integer-matrices-with-...
- 133
4
votes
0
answers
927
views
Generate non-negative linear combinations of non-negative vectors with different supports
(I will not be surprised if this problem has been solved and/or has a trivial solution – I just do not know the right terminology to google for it.)
So the problem is as follows. I have an $m \...
- 325
3
votes
0
answers
463
views
An optimization problem over real symmetric matrices
Given an $n\times s$ matrix $P$ of positive real numbers and $T\geq n$, find (either by a formula or an algorithm) the real symmetric $n\times n$ Z-matrix $A$ which maximizes $\min\limits_{1\leq i<...
- 767
2
votes
1
answer
157
views
Non-negative integer matrix representation of a fusion ring
Context: I am a physics grad student working on topological lines in 2D CFTs.
Let $A$ be a unital based $\mathbb{Z}_{+}$ ring with finite rank (or a Fusion ring) with the basis $B = \{b_1, b_2, \dotsc ...
- 121
2
votes
0
answers
2k
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Sufficient conditions for positive semidefiniteness of block matrix
$\newcommand{\Re}{\mathbb{R}}$I m looking for sufficient conditions that may guarantee positive semidefiniteness (PSD) of a block matrix
$$A = \begin{bmatrix} A_{1,1} & \cdots & A_{1,n} \\ \...
- 121
2
votes
0
answers
327
views
Showing positive stability of a matrix constructed from a positive matrix
A is a positive nonsingular matrix. Let $s>\rho(A)$. We want to show that $B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right)$ is a positive stable matrix, i.e., all eigenvalues of this matrix ...
- 21
2
votes
0
answers
124
views
When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?
We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.
let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times p$...
- 195
2
votes
0
answers
107
views
Polynomials with positive coefficients passing through fixed points/range of Vandermonde matrices
I'll give two equivalent statements of the setup, then give my questions.
Fix integers $M \leq N$ and define the Vandermonde-like matrix $V_{M,N}[i,j] = (1 - \frac{i}{M})^{j-1}$ for $i \in \{1,2,\...
- 21
2
votes
0
answers
352
views
Likelihood convexification
I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form:
$$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log \...
1
vote
0
answers
56
views
Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
- 111
1
vote
0
answers
76
views
Weird transportation polytope
I'm looking to compute extremal points of a weird polytope. This polytope contains all matrices with positive entries $A \in \mathcal M_{n,m}\left(\mathbb R_+\right)$ such that:
every row sum except ...
- 686
1
vote
0
answers
83
views
On the real and finite field rank of a $0/1$ matrix - II
Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$.
Fix a permutation ...
- 13.7k
1
vote
0
answers
86
views
Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix
Lets assume we have the following equation:
$AU=\lambda U \Rightarrow\left[
\begin{array}{c|c|c}
0 &A_{12}&A_{13}\\
\hline
A_{21}& 0& A_{23}\\
\hline
A_{31}&A_{32}&0
\end{...
- 21
1
vote
0
answers
346
views
The problem about the eigenvalues of Metzler matrix in a special form
Let $G$ be a square matrix of the form
$$G=\left[ \begin{array}{cc}
A & B \\
C & 0 \\
\end{array}
\right]$$
with $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \...
- 11
1
vote
0
answers
45
views
Rank Optimization over semi-definite constrains
Let $X$ and $Y$ be finite dimensional Hilbert spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, $L(X)$ and $L(Y)$ be the sets of linear operators of $X$ and $Y$, $\text{Herm}(X)$ and $\text{Herm}(Y)$ be the ...
- 1,493
1
vote
0
answers
266
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Interlacing Suleĭmanova spectra
A set of real numbers $\{\lambda_1, \dots, \lambda_n \}$, $n \geq 1$, is called a Suleĭmanova spectrum if it contains exactly one positive value and $\sum_{i=1}^n \lambda_i \geq 0$. (It is well-known ...
- 1,711
1
vote
0
answers
264
views
A Perron-Frobenius problem
Let $A$ be an irreducible nonnegative matrix with column sums equal to 1.
Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...
- 3,771
1
vote
0
answers
143
views
Bounding the Schur's complement of similiar matrices
Assume the following:
• $L\leq K$
.
• $\Gamma\in M_{K,L}$ is a $L$ rank ${ 0,1} $ matrix, without identical rows or the zeros row.
• $N\in M_{K,K}$ is a diagonal matrix, whose diagonal is a ...
- 295
0
votes
0
answers
224
views
When does a real-valued function of a matrix depend only on eigenvalues?
Let $\mathcal{N}$ be the space of all $n \times n$ matrices that are similar to some nonnegative matrix with zero diagonal and let $f: \mathcal{N} \to \mathbb{R}$ be a continuously differentiable ...
- 1,058