Questions tagged [nonnegative-matrices]

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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 ...
Hannah Cairns's user avatar
26 votes
2 answers
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 ...
Carlo Beenakker's user avatar
25 votes
5 answers
2k views

When is a matrix power nonnegative

The following question came up today during a discussion: Suppose $A$ is an $n \times n$ real matrix. Is there some way to tell whether there exists an integer $q > 0$ such that $A^q$ is ...
Suvrit's user avatar
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21 votes
3 answers
2k views

Sampling from the Birkhoff polytope

The set of $n\times n$ real, nonnegative matrices whose rows and columns sum to one forms the well-known Birkhoff polytope Recently someone asked me if I knew How to sample (in polynomial time) ...
Suvrit's user avatar
  • 28.4k
13 votes
1 answer
1k views

When is a matrix similar to a non-negative matrix?

Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...
J Reichardt's user avatar
12 votes
2 answers
3k views

Eigenvalues of nonnegative integer matrices

Edit I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post: What are the possible eigenvalues of nonnegative integer matrices? Any answer to ...
Brian Rushton's user avatar
12 votes
2 answers
1k views

Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite. My question is: Is there a ...
Omar's user avatar
  • 123
12 votes
3 answers
699 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....
Jeremy Lin's user avatar
11 votes
2 answers
702 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 ...
Surb's user avatar
  • 662
10 votes
2 answers
717 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. ...
vansy's user avatar
  • 143
10 votes
0 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,...
MERTON's user avatar
  • 505
8 votes
4 answers
1k views

Doubly stochastic matrices as squares of entires of unitary matrices

Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a ...
Ben Lerner's user avatar
8 votes
2 answers
692 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 ...
David G. Stork's user avatar
8 votes
1 answer
594 views

Inequalities for Hadamard products of complex symmetric matrices

Consider a complex symmetric matrix $$ C= C_R + i C_I $$ with $C_R,C_I \in \text{Mat}_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C_R$ are all strictly positive. Then, $C$ ...
martin's user avatar
  • 123
8 votes
2 answers
651 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 $\...
kjetil b halvorsen's user avatar
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 ...
Mahdi's user avatar
  • 1,991
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 & ...
Hadrien's user avatar
  • 181
7 votes
1 answer
4k views

How much can a diagonal matrix change the eigenvalues of a symmetric matrix?

Suppose that we have a symmetric matrix ${\bf S}$ with eigenvalue decomposition ${\bf S} = {\bf Q}{\bf \Lambda}{\bf Q}^T$. Assume that we have two diagonal matrices ${\bf D}_1$ and ${\bf D}_2$ that ...
Anadim's user avatar
  • 449
7 votes
2 answers
735 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: $$ \operatorname{tr} \left( A e^{B+C} \right) \leq \...
nikka's user avatar
  • 375
7 votes
1 answer
439 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 $\...
valle's user avatar
  • 864
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\...
user avatar
7 votes
1 answer
382 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 $col(A)$. ...
bodhisat's user avatar
6 votes
2 answers
1k views

Extreme points of transportation polytope

I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ...
Memming's user avatar
  • 291
6 votes
2 answers
437 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\\ \...
pre-kidney's user avatar
  • 1,289
6 votes
1 answer
886 views

A matrix inequality involving the Hilbert-Schmidt norm

This question comes from a problem in PDEs on which I'm currently working. Let $a$ be a $3\times 3$ matrix, real symmetric and positive definite. Denote with $\|a\|^2 _ 2=\sum a_{ij}^2$ the square of ...
Piero D'Ancona's user avatar
6 votes
3 answers
524 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 = \...
Ariel's user avatar
  • 63
6 votes
2 answers
4k views

similarity transformation into symmetric matrices

I want to determine some structures of matrices that can be transformed into a symmetric matrices using similarity transformation, i.e., $B=T^{-1}AT$ where $T$ is the similarity transformation ...
Xiao Junhui's user avatar
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_+-...
Turbo's user avatar
  • 13.7k
5 votes
2 answers
1k views

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
Hans's user avatar
  • 2,169
5 votes
2 answers
274 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 ...
User8976's user avatar
  • 199
5 votes
4 answers
759 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). ...
Pietro Paparella's user avatar
5 votes
1 answer
589 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_{...
baptiste's user avatar
  • 123
5 votes
1 answer
407 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&...
Amin235's user avatar
  • 325
5 votes
1 answer
609 views

Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
Suvrit's user avatar
  • 28.4k
5 votes
0 answers
424 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 ...
loup blanc's user avatar
  • 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-...
Navin K.'s user avatar
  • 133
4 votes
1 answer
541 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, $I_{n*n}$...
Phani Raj's user avatar
  • 143
4 votes
2 answers
396 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 ...
Pietro Paparella's user avatar
4 votes
1 answer
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? ...
Turbo's user avatar
  • 13.7k
4 votes
2 answers
185 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 ...
groupoid's user avatar
  • 580
4 votes
1 answer
423 views

non negative solution of the matrix equation $A^T U A = U - C$ if C is non-negative

Given $A$ a matrix with spectral radius smaller than 1 and a matrix $C$ symmetric. It can be shown that $U=\sum_{k=0}^\infty (A^T)^k C A^k$ converges, is symmetric and is the solution of the equation ...
Giovanni's user avatar
4 votes
1 answer
214 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 ...
subshift's user avatar
  • 1,110
4 votes
1 answer
291 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\...
Turbo's user avatar
  • 13.7k
4 votes
1 answer
405 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 ...
nick.schachter's user avatar
4 votes
1 answer
429 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&...
Mohammad Khosravi's user avatar
4 votes
0 answers
923 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 \...
Yauhen Yakimenka's user avatar
3 votes
1 answer
3k views

Non symmetric matrices with real eigenvalues

Consider the following block matrix $A=\pmatrix{A_1 & A_2\cr kA_2^\top & A_3}$ where $A_1$ is a symmetric matrix, $A_3$ is diagonal matrix and all entries of $A$ are real and non-negative. ...
Jamil Tau's user avatar
3 votes
1 answer
179 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 = \...
rhombidodecahedron's user avatar
3 votes
1 answer
902 views

numerical range of a column-zero-sum matrix

I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
Delio Mugnolo's user avatar
3 votes
2 answers
1k 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 &...
ram's user avatar
  • 31