# Questions tagged [nonnegative-matrices]

A non-negative matrix is one whose entries are non-negative.

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### stability of rank one 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 ...
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### Eigenvalues of the product of a stochastic matrix and a doubly stochastic matrix

I am playing with compositions of finite state Markov chains and have this product matrix $P=AB$, where $A$ is a stochastic matrix (rows sum to 1) and $B$ is a doubly stochastic matrix. Moreover, $B$ ...
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1 vote
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### Optimal transition matrix that minimizes mixing time

For a finite state space, say I have a transition matrix $P$ such that it is irreducible and aperiodic. The stationary distribution is $\pi$. I wonder if there's any literature on how to pick an extra ...
• 151
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### Change of basis to a nonnegative matrix

Let $A$ be an arbitrary symmetric, square matrix. I would like to use the Perron-Frobenius theorem, but I cannot do that directly since $A$ is not necessarily a non-negative matrix. So I had the ...
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• 1,504
1 vote
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### How to understand the KL-divergence between two non-negative matrices

I am reading Non-negative Matrix algorithm using KL-divergence as metric. The KL-divergence is known as $D(P,Q)=\sum_i P(i)log\frac{P(i)}{Q(i)}$ for discrete distribution. However, the KL-divergence ...
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### Does this function preserve the order of the nonnegative matrices

Suppose A and B are nonnegative matrices, that is, $A\geq 0$ and $B\geq 0$. Moreover, suppose $\|A\|_{\infty}<1, \|B\|_{\infty}<1$ and $A\leq B$. For $f(x)=x(2-x)$, we know that $f(x)$ is ...
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### How to prove the irreducible non-negative matrix has positive geometric sum?

Here is a question: A matrix $A=(A_{ij})_{n\times n}$ is called irreducible iff for every pairs $(i,j)$ we have $(A^k)_{ij}>0$ for some $1\leq k\leq n-1$. How to prove that matrix $A$ is ...
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### A matrix $A$ is $K$-irreducible if and only if no eigenvector lies on $\partial K$.

I'm studying the properties of nonnegative matrices, and I encountered a theorem for which I can not understand its proof. The theorem can be found in "Nonnegative matrices in the Mathematical ...
• 1,669
1 vote
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### Criterion for non-negative matrix to have strictly real eigenvalues

Question: I am wondering whether there exists a sufficient and necessary condition that has to be obeyed by a non-negative $n\times n$ matrix $A=\{a_{ij}\}\in \mathbb{R}^{n \times n}$, which ...
1 vote
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### Rational left eigenvector in Perron-Frobenius theorem with nonnegative integer matrix

On pages 1133-1135 of Joel Friedman's On the road coloring problem, (PAMS 1990), we have an $n \times n$ primitive integer matrix $A$ with all row sums equal to $d \geq 2$. Clearly the right Perron ...
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### References on nonnegative matrices

I just started researching on non-negative matrices and their properties. Any suggestions on papers or sources for the theory and applications (ideally recent ones) of such matrices? I am currently ...
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1 vote
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### How to manually find NMF for a given matrix (of small size)

I would like to know (if possible) how does one find exact the non-negative matrix factorization ($AW$) of a given matrix $M$ of a small size? I know the problem is NP-hard. The method I know is by ...
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1 vote
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### non negative matrices and spectrum

Let $r$ be the Perron Frobenius eigenvalue of a non negative matrix $A$. It seems that it's true that if $r, -r \in Spec(A)$ then $-A = S\cdot A \cdot S^{-1}$ with $S$ a diagonal matrix with $\pm 1$ , ...
1 vote
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### If A and B are Hermitian Matrices, $\text{tr}(ABAB) \geq 0$.

Can anybody help me with this problem. If $A$ and $B$ are Hermitian Matrices, then $\text{tr}(ABAB) \geq 0$. It is easy to show that the trace is a real number, but I cannot prove that it must be non-...
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