A non-negative matrix is one whose entries are non-negative.
A non-negative matrix $M$ is such that $m_{ij}\geq 0$. The Perron-Frobenius theorem asserts the existence of a unique largest positive eigenvalue of non-negative matrices.
Questions tagged [nonnegative-matrices]
95 questions
36
votes
2 answers
Finding path-lengths by the power of Adjacency matrix of an undirected graph
I knew from Mark Newman's book - Networks: An Introduction (Page 137, Eq: 6.31) that, if $A$ is the adjacency matrix of a graph, then $ij$'th entry of $A^k$ will give me the number of $k$-length paths connecting the vertices $i$ and $j$. This…
kada mati
- 463
9
votes
1 answer
How to tell if a directed graph is acyclic from the adjacency matrix?
Suppose you have an adjacency matrix $A$ for a directed graph $G=\{V,E\}$, so $A_{ij} = 1$ if $V_i\rightarrow V_j \in E$, and $A_{ij}=0$ otherwise. Many properties of the graph can be derived from this adjacency matrix. For instance, $(A^n)_{ij}$…
David Pfau
- 1,149
9
votes
1 answer
Nonnegative orthogonal matrices
Assume that $A \in \mathbb{R}^{n \times n}$ has nonnegative entries and $AA^T = I_n$ where $I_n$ is the identity matrix. Is it true that $A$ should be a permutation matrix?
EDIT: I seem to have a proof for doubly stochastic matrices based on the…
passerby51
- 4,280
6
votes
1 answer
Proofs of Karpelevich's results about eigenvalues of nonnegative matrices.
Are there any books or papers written in English that contain proofs of the results obtained in the following paper?
F.I. Karpelevich, On the characteristic roots of matrices with nonnegative elements, Izv. Akad. Nauk SSSR Ker. Mat., 1951, v.15,…
user1551
- 149,263
6
votes
2 answers
How to convert any non-negative matrix into a doubly stochastic matrix?
Given a non-negative real matrix $A \in \Bbb R_+^{m \times n}$, how do I convert it to a doubly stochastic matrix (each row and column sums to $1$)
$$\sum_{j=1}^n A_{ij}= 1, \qquad \forall i = 1, \dots, m \tag{row sum}$$
$$\sum_{i=1}^m A_{ij}= 1,…
Learner
- 2,796
5
votes
3 answers
Intuition of $D\leftarrow XC^{T}\text{diag}(C1_n)^{-1}$ update rule in matrix factorization
I am reading this paper where they use Matrix Factorization over Attention mechanism in their Hamburger model. In section 2.2.2 they say,
Vector Quantization (VQ) (Gray & Neuhoff, 1998), a classic data compression algorithm, can be formulated as an…
WhyMeasureTheory
- 895
5
votes
0 answers
Maximum element of Perron vector
Suppose $A$ is an entrywise nonnegative matrix that is symmetric, irreducible and has a zero diagonal. By Perron-Frobenius theorem, the spectral radius $\rho(A)$ is an eigenvalue and $A$ has, up to scaling, a unique entrywise positive eigenvector…
user1551
- 149,263
4
votes
0 answers
Dimension of the fixed space of stochastic matrices (reference request)
Let $A \in \mathbb{R}^{d \times d}$ be row stochastic (i.e., all entries of $A$ are $\ge 0$, and each row sums up to $1$.
Let $G(A)$ denote the directed graph that is associated to $A$, i.e., $G(A)$ has vertices $\{1, \dots, n\}$, and there is an…
Jochen Glueck
- 909
4
votes
1 answer
Product of a primitive matrix and its transpose.
Is it true that if $A$ is a nonnegative primitive matrix, then $AA^T$ is also primitive? Obviously $A^T$ is primitive but in general product of primitive matrices is not primitive.
Any hint?
aleio1
- 1,065
3
votes
1 answer
Eigenvectors of reversible stochastic matrix
Let $P$ be the stochastic matrix associated with a reversible Markov chain and $\pi$ its stationary distribution. Then it can be seen that $DPD^{-1}$ where $D=\text{diag}(\sqrt{\pi(x)})$ is symmetric and hence its eigenvalues are real and its…
Navid Rashidian
- 375
3
votes
1 answer
Stability of rank-$1$ perturbation of stochastic matrix
Let $I$ be the identity matrix and $P$ be an irreducible $n$ by $n$ row stochastic matrix. Let $d$ be a stochastic (column) vector and $e$ be an all one (column) vector. Let $t > 0$ be a real number. Define $$A(t) = \rho(P)I - P + t de^\top,$$ where…
AlphaRL
- 175
3
votes
0 answers
Spectral Radius of the Convex Combination of Identity Matrix and a Nonnegative Matrix
Let $D = \operatorname{diag}(d_1, d_2, \ldots, d_n)$ with each of its diagnal entry $d_i \in (0,1)$. $B$ is a non-negative matrix.
Consider the following matrix
$$
A = DI + (I-D)B,
$$
where $I$ is the identity matrix. Each row of $A$ is a linear…
maphado fan
- 43
3
votes
0 answers
Can we determine whether a symmetric matrix appended by one column/row maintains the Perron-Frobenius property?
Assume we are given a symmetric,non-negative, irreducible matrix $A \in \mathbb{R}^{n \times n}$.
Then by the Perron-Frobenius theorem, there exists an eigenvector $v \in \mathbb{R}^n$ for the eigenvalue $\sigma(A)>0$ and the eigenvector $v$ is…
a_student
- 1,514
3
votes
1 answer
Perron-Frobenius Theorem poof by Brouwer fixed point
Could you suggest me a book where I can find a proof of Perron-Frobenius theorem (especially for nonnegative matrices) based on a Brouwer fixed point theorem?
3
votes
0 answers
Rank of non-negative matrices
Are there some easy ways to verify conditions on a nonnegative matrix with $0$ diagonal which ensure that it is conditionally negative definite (i.e., $x^TAx \le 0$ for all $x$ with $x^T\mathbf{1}=0$ and equality holds if and only if $x=0$)?…
