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Questions tagged [nonnegative-matrices]

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0
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1answer
26 views

Non negative irreducible matrix times a positive matrix

Let $A$ be a real non negative irreducible matrix, meaning that $(a_{ij})\geq 0$ I know from a theorem in Minc's book that $(I+A)^{n-1}>0$. It is also stated in the book that $B=(I+A)^{n-1}.A>...
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0answers
24 views

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 ...
0
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1answer
12 views

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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0answers
14 views

When is the spectral radius of a submatrix strictly bounded by the original matrix.

Let $A$ be a matrix in $\mathbb R^{n\times n}$. Define an orthonormal basis $\{ w_1,w_2,\dots,w_n \}$ of $\mathbb R^n$. Consider a subset of $m$ basis vectors, say $\{ w_j | j\in \mathcal J\}$ and $\...
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0answers
29 views

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 ...
0
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1answer
16 views

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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0answers
31 views

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 ...
0
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0answers
27 views

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 ...
2
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1answer
27 views

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 ...
1
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0answers
23 views

Convergence of Matrix product (not memoryless)

I try to analyze the following iterative matrix product: $x_{i+1} = A_{i} \cdot x_{i}$. The matrices $A_i$ are defined as follows: $A_i = \begin{bmatrix} \frac{1}{2\sqrt{x_{i,1}^2 + y_{i,1}^2}} &...
2
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2answers
68 views

Positive matrices are diagonalizable in $\mathbb C$?

Let $A$ be a square matrix with real positive entries. Is it true that it is diagonalizable in $\mathbb C$? My guess is that it is false, but the counterexample can't be a $2\times 2$ matrix, since ...
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0answers
31 views

Matrix Factorization with Binary/Non-negative factor constraints

I am looking for a constrained factorization solver that allows for matrix factorization subject to constraints that one of the factors is binary, and the other is non-negative. To put it formally, ...
1
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1answer
77 views

Comparing spectral radii of two non-negative matrices

Let $A$ and $B$ be $n \times n$ non-negative matrices with spectral radii $\lambda_A$ and $\lambda_B$, respectively. Suppose that $A_{ij} \leq B_{ij}$ for all $i,j$. Then, $$\lambda_A \leq \lambda_B$$ ...
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0answers
43 views

If A is primitive, is AA' primitive? [duplicate]

Let $A$ be a primitive matrix (a square nonnegative matrix some power of which is positive). Is $AA^T$ necessarily primitive?
0
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1answer
95 views

Eigenvectors to the largest eigenvalue $\lambda_1$ of $A\geq 0$ are non-zero only at the end of the longest chains when $\lambda_1=0$?

Let $n>0$ and $A\in M_{n\times n}(\{0,1\})$. Suppose the largest eigenvalue $\lambda_{1}=0$ vanishes. Then there is no closed path in the graph, only chains. I have read the claim that only the end ...
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0answers
166 views

What is the Perron-Frobenius theorem for non-negative matrices?

Let $M\succeq 0$ (i.e. $M_{ij}\geq 0$ for all $i,j$). No further conditions on $M$ such as irreducibility, aperiodicity, or what not. What is the formulation of the theorem in this case? I believe ...
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0answers
18 views

How does the geometrical definition of irreduciblity implies the combinotarial definition?

Recently I have come across the geometrical definition of irreducible matrix, as goes follow. A matrix in $\pi(K)$ is $K-$irreducible if the only faces of $K$ that it leaves invariant are $\{0\}$ ...
1
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1answer
158 views

On vectors shifted by $1$?

Denote $u\leq v$ to imply vector $u$ lower bounds all coordinates of $v$. Given a rank $n$ matrix $M\in\mathbb Q^{n\times n}$ with non-negative entries and with $M_i\not\leq M_j$ at every $i,j\in\{1,...
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0answers
21 views

Estimating a matrix from another matrix

The problem is the non- negative matrix factorization of a matrix. Let me explain my problem I have an original matrix $A=\begin{bmatrix} 0.248437 &0.25198098 & 0.25396825 & 0.25077881\\ 0....
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0answers
32 views

NMF factorization to get the desired matrix

I am working on an encryption algorithm for images, and am stuck in the following factorization. Let $A$ be a positive, invertible, non-sparse random matrix. Let $S$ be my image matrix that is ...
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0answers
86 views

Construction of a non-negative matrix

I am designing an algorithm, in which an intermediate step is to create a $n\times n$ invertible non-negative matrix. The idea that I thought was to choose a random generator $P$ to generate $n^2$ ...
3
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1answer
114 views

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?
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0answers
39 views

Fully indecomposable matrix is primitive

A non negative matrix $A\in\mathbb{R}^n$ is said fully indecomposable if it doesn't exist permutation matrices $P,Q$ such that $$PAQ=\begin{bmatrix}X &0\\Y & Z\end{bmatrix}$$ where $X,Z$ are ...
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0answers
24 views

Irreducible matrix

Is $$ \left[ \begin{array}{lll} (0)_{4 \times 4} & (1) & (1)\\ (1) & (0)_{4 \times 4 } & (1)\\ (1) & (1) & (0)_{12 \times 12} \end{array} \right] $$ an irreducible matrix?
1
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0answers
54 views

Unitary transformation to non-negative matrix.

I am looking to perform a unitary transformation $U$ on a matrix $S$ such that \begin{equation} Q = USU^\dagger \end{equation} is an element-wise non-negative matrix. I would like to know whether ...
1
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0answers
75 views

Multiplicity in the largest eigenvalue of a non-negative irreducible symmetric matrix (primitive and imprimitive matrix))

We know from the Perron-Frobenius theorem that for a non-negative matrix, $A$, (symmetric or asymmetric) the largest eigenvalue is simple and it has a positive eigenvector if the matrix is irreducible ...
2
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1answer
59 views

Why $K$ is solid if its reproducing?

Let $K$ be a closed convex cone. Then $K$ is solid if and only if $K$ is reproducing. Hint: If $K$ is a convex cone then $K-K$ has a nonempty interior. $K-K$ is the minimal subspace containing $K$. ...
12
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2answers
19k views

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 ...
6
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1answer
2k views

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 ...