Questions tagged [nonnegative-matrices]

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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\...
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17 votes
1 answer
639 views

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 ...
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8 votes
2 answers
315 views

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 $\...
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1 vote
1 answer
182 views

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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1 vote
0 answers
68 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 ...
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  • 621
1 vote
0 answers
55 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 ...
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  • 12.8k
4 votes
1 answer
164 views

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\...
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  • 12.8k
0 votes
1 answer
101 views

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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5 votes
0 answers
343 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 ...
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  • 2,850
4 votes
2 answers
231 views

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 ...
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8 votes
0 answers
340 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 & ...
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  • 181
1 vote
0 answers
85 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{...
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5 votes
1 answer
345 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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  • 315
1 vote
1 answer
230 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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7 votes
1 answer
336 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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1 vote
0 answers
290 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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  • 11
8 votes
1 answer
279 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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  • 1,961
5 votes
2 answers
271 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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  • 199
5 votes
1 answer
451 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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  • 113
1 vote
0 answers
43 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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  • 1,265
44 votes
3 answers
3k 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: https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
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6 votes
0 answers
97 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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  • 12.8k
4 votes
1 answer
273 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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  • 12.8k
4 votes
0 answers
758 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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3 votes
1 answer
748 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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2 votes
0 answers
992 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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1 vote
1 answer
168 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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4 votes
1 answer
340 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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6 votes
1 answer
340 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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5 votes
4 answers
570 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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1 vote
0 answers
265 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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2 votes
1 answer
178 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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3 votes
2 answers
350 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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12 votes
3 answers
650 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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2 votes
1 answer
183 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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  • 23
4 votes
2 answers
143 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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  • 590
4 votes
1 answer
536 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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  • 143
1 vote
1 answer
183 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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2 votes
0 answers
292 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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  • 21
2 votes
1 answer
278 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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  • 1,071
9 votes
0 answers
391 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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  • 455
2 votes
3 answers
213 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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  • 21
8 votes
2 answers
612 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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2 votes
0 answers
122 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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11 votes
2 answers
629 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 $a_{ij} \ge 0$ for all $1 \le i,j \le n$. Chapter 8 ...
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  • 632
6 votes
3 answers
474 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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  • 63
1 vote
0 answers
258 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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2 votes
0 answers
98 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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  • 21
7 votes
1 answer
1k 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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5 votes
0 answers
223 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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