# Questions tagged [nonnegative-matrices]

The nonnegative-matrices tag has no usage guidance.

84
questions

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### 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-...

2
votes

1
answer

151
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### 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 ...

1
vote

1
answer

155
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### Second order matrix differential equation in the space of symmetric positive definite matrices

In the construction of interpolations in the space of Gaussian measures, I encountered a second order matrix differential equation in the set of symmetric positive definite matrices $\mathbb{S}_+^d\...

26
votes

2
answers

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### Uniquely reconstruct a matrix $M$ from its inverse $M^{-1}$ if $n$ elements of $M^{-1}$ are unknown and $n$ elements of $M$ are given

This question was motivated by a recent MO post. You know $n$ elements of the $N\times N$ matrix $M$ and you do not know $n$ elements of the inverse $M^{-1}$ (but you know the other $N^2-n$ elements ...

8
votes

2
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636
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### Finding a matrix from its diagonal and the off-diagonal elements of its inverse?

This question comes from https://stats.stackexchange.com/questions/457375/recover-full-covariance-matrix-from-covariance-diagonal-and-precision-off-diagon where it have not found answers. So, let $\...

1
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1
answer

217
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### Growth rates of ‘sub-traces’ of matrices

Consider an aperiodic, non-negative and non-zero $m\times m$ matrix $A$. Here aperiodic means that there exists an integer $n$ such that every entry of $A^n$ is strictly positive. By the Perron–...

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0
answers

76
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### 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 ...

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0
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82
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### 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 ...

4
votes

1
answer

284
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### On the real and finite field rank of a $0/1$ matrix - I

Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$.
Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by $m\...

0
votes

1
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110
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### Matrix iteration for non-negative matrices. Does it converge to some eigenvector?

Let $A$ be a non-negative (entrywise) matrix such that $A(1,1)>0$. Set $u=(1,0,0,...,0)^T$. Is it always true that there exists a non-negative eigenvector $v$ of $A$ such that $\lim_{n\rightarrow\...

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0
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423
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### 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 ...

4
votes

2
answers

373
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### Frobenius normal form of a doubly stochastic matrix

If $A \in M_n(\mathbb{C})$, then $A$ is called reducible if there is a permuation matrix $P$ such that
$$
P^\top A P =
\begin{bmatrix}
A_{11} & A_{12} \\
0 & A_{22}
\end{bmatrix}, $$
in ...

8
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0
answers

394
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### 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 & ...

1
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0
answers

86
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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{...

5
votes

1
answer

406
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### 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&...

1
vote

1
answer

426
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 ...

7
votes

1
answer

381
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 $col(A)$.
...

1
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0
answers

344
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### 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 \...

8
votes

1
answer

287
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 ...

5
votes

2
answers

273
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### 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 ...

5
votes

1
answer

579
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_{...

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 ...

47
votes

3
answers

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### 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:
https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf
I've written an outline of it below ...

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_+-...

4
votes

1
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289
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### 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?
...

4
votes

0
answers

917
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### 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 \...

3
votes

1
answer

968
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.

2
votes

0
answers

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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} \\ \...

1
vote

1
answer

214
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### 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 ...

4
votes

1
answer

402
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### 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 ...

7
votes

1
answer

436
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 $\...

5
votes

4
answers

746
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). ...

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 ...

2
votes

1
answer

181
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### 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 ...

3
votes

2
answers

505
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 ...

12
votes

3
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697
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### 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....

2
votes

1
answer

201
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### 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 ...

4
votes

2
answers

184
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### 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 ...

4
votes

1
answer

539
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### 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, $I_{n*n}$...

1
vote

1
answer

194
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### 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 ...

2
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0
answers

325
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### 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 ...

2
votes

1
answer

366
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### 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 ...

10
votes

0
answers

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### 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,...

8
votes

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689
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### 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 ...

2
votes

0
answers

124
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### 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 $p\times p$...

11
votes

2
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691
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### 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 $a_{ij} \ge 0$ for all $1 \le i,j \le n$.
Chapter 8 ...

6
votes

3
answers

520
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### 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 = \...

1
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0
answers

262
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### 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 ...

2
votes

0
answers

107
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### 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,\...

7
votes

1
answer

1k
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### 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\...