# Questions tagged [nonnegative-matrices]

The nonnegative-matrices tag has no usage guidance.

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### 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{...

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**1**answer

275 views

### Determining the primitive order of a binary matrix

Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows
$$
{\bf A}_n=\left(
\begin{array}{c}
0&0&\cdots&0&0&0&0&1&1\\
0&0&\cdots&0&0&...

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**1**answer

145 views

### Is there a way to turn a non convex set to a convex one? [closed]

Perhaps the question is rather vague, but if we are given a non-convex set S, can we construct some invertible mapping f so that f(x) becomes a convex set?
In my problem , I am given a set of matrix ...

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**1**answer

232 views

### Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?

Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:
The all-one vector $j$ is contained in the conic hull of $col(A)$.
...

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154 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 \...

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votes

**1**answer

256 views

### Distance from nonnegativity of some orthonormal vectors

Suppose that $1 < k < n$. Does there exist a constant $\beta > 0$, such that for every $k$ orthonormal vectors $f_1,\ldots,f_k \in \mathbb R^n$,
there exist $k$ orthonormal vectors with ...

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**2**answers

258 views

### An equality relation for complex numbers off the nonnegative real axis [closed]

For every complex number $z$ off the nonnegative real axis there exist positive numbers $p_0,... ,p_n$ such that $\sum_{i=0}^n p_iz^i = 0$.
Finding difficulty in proceeding with the problem. Need ...

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votes

**1**answer

227 views

### Nonnegative matrices and singular values

I would like to prove (or prove it is not true with a counter example) the following result:
Let $A$, $B$ be two squares matrices of size $n\times n$ with positive entries.
If $A \leq B$, then $\sum_{...

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41 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 ...

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1k views

### Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation:
http://www.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf
I've written an outline of it below ...

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91 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_+-...

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**1**answer

256 views

### What is this matrix decomposition called and does it exist always?

Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ 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 $r$ such that $M=M_+-M_-$ holds?
...

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356 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 \...

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**1**answer

292 views

### The largest eigenvalue of a binary matrix with specific density

I would like to find the largest eigenvalue of an $n \times n$ binary matrix of density $p$, i.e., with $p n^{2}$ ones and $(1-p) n^{2}$ zeros. Any idea or reference is welcome.

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230 views

### 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} \\ \...

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vote

**1**answer

80 views

### zero patterns of M-matrices and their invereses

Suppose that the matrix $M$ has non-positive off-diagonal elements. The matrix $M$ is
said to be an M-matrix if $M$ is non-singular and each entry of $C:=M^{−1}$
is non-negative.
I think I've proved ...

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votes

**1**answer

156 views

### Proving convergence of modified ALS for non-negative matrix factorization

The general Non-Negative Matrix Factorization (NMF) problem asks that for given a matrix $X \in \mathbb R^{m,n}_{0+}$ (with this notation meaning a matrix containing real numbers greater than or equal ...

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**1**answer

228 views

### “Unimodality” of the positive eigenvector of a non-negative irreducible matrix?

Consider an eigenvalue / eigenvector problem for a matrix $A$ that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):
$$\sum_j A_{ij} x_j = \lambda x_i$$
Here $\...

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votes

**4**answers

326 views

### Expected value of the spectral radius of a random nonnegative matrix

I have two questions, the second of which is related to this question posed by Denis Serre.
Let $X$ be a random variable and suppose that $Y=|X|$ (e.g., $Y$ could be the folded normal distribution). ...

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262 views

### 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 ...

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votes

**1**answer

153 views

### Perturbation of linear system of equations: Is the solution still non-negative?

Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a matrix such that
$a_{ij} \ge 0$ for all $i,j = 1, \dots, n,$ and
$A$ is positive definite.
Let $I$ be the identity matrix, and $\pmb{1}$ the vector ...

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votes

**2**answers

212 views

### Number of $\{0,1\}$ matrices with distinct rows and distinct columns

How many $M\in\{0,1\}^{r\times c}$ are there such that each row and each column of $M$ is distinct?
How many classes of matrices in $\{0,1\}^{r\times c}$ up to permutation equivalence are there such ...

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**3**answers

572 views

### How to find Suleimanova's work on the Nonnegative Inverse Eigenvalue Problem?

Many papers cite the work of Suleimanova when studying inverse eigenvalue problems - in particular, the nonnegative inverse eigenvalue problem (NIEP). However, I cannot seem to find her work anywhere....

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votes

**1**answer

127 views

### Perron-Frobenius for a “maximum” of matrices

Consider a set of positive matrices $(P_k)_{k\in K}$ in $\mathbb R^{p\times p}_{++}$ ($P_k$ is positive in the sense that all entries of $P_k$ are positive).
Let $X_0\in\mathbb R_{++}^p$ and define ...

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97 views

### Requesting reference for result from linear algebra on Schur complements of M-matrices

In my research in linear algebra, I have come across a useful result stating that the Schur complement of a principal non-singular submatrix of an M-matrix is also an M-matrix, but I have never found ...

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525 views

### No arbitrary product of matrices has eigenvalue 1?

Consider the matrix $D$, adjacency matrix of an undirected graph $G$ on $n$ vertices, with the notation that $d_{i,i}=0,\forall i$.
The matrices $A_i$ are constructed from Identity matrices, $I_{n*n}$...

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111 views

### nonnegative solution of nonhomogeneous under-determined linear system of equations

For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such ...

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202 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 ...

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**1**answer

180 views

### Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...

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364 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,...

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168 views

### Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists.
Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory).
Preferably the ...

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436 views

### Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries

This is a more carefully worded version of this question, here tailored to professional mathematicians.
Consider a matrix ${\bf A}\in{\bf M}_{n\times n}({\mathbb R})$ with possibly positive, negative ...

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110 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 $p\times p$...

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459 views

### Encyclopedia of properties of nonnegative matrices

I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $A_{i,j}\geq 0$ for all $i,j$.
Chapter 8 in Matrix ...

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**3**answers

379 views

### Product and convex combination of two stochastic matrices

Let $K_1$ and $K_2$ be two $N \times N$ stochastic matrices (hence non-negative and rows adding up to one) with zeros on the diagonal. If $\alpha \in (0,1)$, is it possible to have
$$K_1 K_2 = \...

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248 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 ...

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89 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,\...

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votes

**1**answer

778 views

### Determinant of some covariance matrix (Gaussian kernel process)

Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - x_j\...

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195 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-...

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301 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 \...

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172 views

### Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$.
Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...

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**1**answer

87 views

### Are certain normal matrices circulant? (Part 2)

Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $
A^TA=\begin{pmatrix}
a & b \\
b & \ddots
\end{pmatrix}$, for $b>0$.
As a user observed in the solution of Part 1 ...

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votes

**2**answers

355 views

### Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that
$$
A^TA=\begin{pmatrix}
a & b & \cdots & b\\
b & a & \ddots & \vdots\\
\...

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548 views

### Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
...

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163 views

### Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

The nonnegative matrix
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = \...

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votes

**1**answer

175 views

### An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases:
$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$
Note that if $A=I$ then ...

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**0**answers

122 views

### A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...

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**2**answers

715 views

### Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements

is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form :
\begin{pmatrix}
1 & b & 0 & ... & 0 \\\
b & 2 &...

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347 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<...

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**1**answer

394 views

### Finding all local maximum points of a function?

Let ${\boldsymbol \theta}=(\theta_1,\theta_2,\ldots,\theta_n) \in{\mathbb T}^n$ and $P:{\mathbb T}^n\rightarrow {\mathbb R}$ be a function defined on $n$-torus as
$$
P({\boldsymbol \theta}) = \sum_{i&...