# Questions tagged [nonnegative-matrices]

The tag has no usage guidance.

84 questions
Filter by
Sorted by
Tagged with
1 vote
49 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
151 views

• 1,081
1k 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 ...
• 176k
636 views

• 13.6k
110 views

• 11
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 ...
• 1,991
273 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 ...
• 199
579 views

• 13.6k
289 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? ...
• 13.6k
(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}$zeros. Any idea or reference is welcome. • 159 2 votes 0 answers 2k views ### Sufficient conditions for positive semidefiniteness of block matrix$\newcommand{\Re}{\mathbb{R}}$I m looking for sufficient conditions that may guarantee positive semidefiniteness (PSD) of a block matrix $$A = \begin{bmatrix} A_{1,1} & \cdots & A_{1,n} \\ \... • 121 1 vote 1 answer 214 views ### zero patterns of M-matrices and their invereses Suppose that the matrix M has non-positive off-diagonal elements. The matrix M is said to be an M-matrix if M is non-singular and each entry of C:=M^{−1} is non-negative. I think I've proved ... 4 votes 1 answer 402 views ### Proving convergence of modified ALS for non-negative matrix factorization The general Non-Negative Matrix Factorization (NMF) problem asks that for given a matrix X \in \mathbb R^{m,n}_{0+} (with this notation meaning a matrix containing real numbers greater than or equal ... 7 votes 1 answer 436 views ### "Unimodality" of the positive eigenvector of a non-negative irreducible matrix? Consider an eigenvalue / eigenvector problem for a matrix A that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):$$\sum_j A_{ij} x_j = \lambda x_i$$Here \... • 864 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). ... • 1,711 1 vote 0 answers 266 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 ... • 1,711 2 votes 1 answer 181 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 ... 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 answers 697 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.... • 121 2 votes 1 answer 201 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 ... • 23 4 votes 2 answers 184 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 ... • 580 4 votes 1 answer 539 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}... • 143 1 vote 1 answer 194 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 ... • 13 2 votes 0 answers 325 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 1 answer 366 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 ... • 1,071 10 votes 0 answers 400 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 2 answers 689 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 ... • 2,518 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 11 votes 2 answers 691 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 ... • 652 6 votes 3 answers 520 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 = \... • 63 1 vote 0 answers 262 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 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,\...
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\...