# Questions tagged [nonnegative-matrices]

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

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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 $\pm 1$ , ...
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### Does the nuclear norm define a kernel?

Consider a function $k : \mathbb{R}^{m \times n} \times \mathbb{R}^{m \times n} \to \mathbb{R}_+$, defined by $k(X, Y) =\Vert X^T Y \Vert_*$ where $\Vert \cdot \Vert_*$ denotes the nuclear norm (sum ...
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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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### Invertible matrix with nonnegative entries [duplicate]

Let $A \in \text{GL}(n,\mathbb R)$ with all nonnegative entries such that $A^{-1}$ also has all nonnegative entries. How many nonzero entries does $A$ have, what are their locations, and what are the ...
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### Special non-negative matrix decomposition

Assume we have the following decomposition: $M=AWA^T$ Where: $M$ is non-negative symmetric (i.e. $M_{i,j}\ge{0}$ and $M_{i,j}=M_{j,i}$) $A$ is right stochastic (entries are non-negative and rows ...
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### Prove $(E+A)^{n-1}$ is a nonnegative matrix with positive elements

Suppose $A$ is an $n\times n$ irreducible nonnegative matrix, prove that $(E+A)^{n-1}$ is a nonnegative matrix with positive elements. I think that this excise can be proved by definition. So I tried ...
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### What are the features extracted in non-negative matrix factorization?

The application of NMF is related to extracting features in original data present as a matrix. An important problem when utilizing NMF for feature extraction is the choice of rank r. However I can't ...
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### Non-Negative Vs Positive Semi Definite

A matrix is PSD if $$\langle Ax, x\rangle \ge 0, \forall x \in H$$ Where, H is a hilbert space and A is a mapping $H \rightarrow H$. Is it the same as being Non-negative? I couldn't seem to find a ...
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### Prove the inverse of a nonnegative matrix is nonnegative

Defintion of a nonnegative matrix: Symmetrical matrix $A: n \times n$ is non-negatively defined when $A > 0$ or $A ≥ 0$ We have to prove the following: If $A$ is defined as a nonnegative matrix, ...
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### Proving that a matrix is nonnegative if its powers are nonnegative

I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
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### If $A$ is nonnegative definite then so is $BAB'$

Let $A$ be an $m$x$m$ nonnegative definite matrix, while $B$ is an $n$x$m$ matrix. Show that $BAB'$ is a nonnegative definite matrix. I know by definition that a nonnegative definite matrix has the ...
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### Number of weak components in powers of imprimitive digraphs

Given any strongly connected digraph $G$ and any $n\in\mathbb{N}$ if we let $d(G)$ be the greatest common factor of the lengths of all the directed cycles in $G$ then does the $n^{\text{th}}$ power ...