Questions tagged [nonnegative-matrices]
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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 ...
12
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2
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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 ...
21
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3
answers
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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) ...
7
votes
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answer
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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 ...
26
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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 ...
8
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2
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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 $\...
6
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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\\
\...
4
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1
answer
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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\...
4
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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?
...
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Properties of eigenvalues of general nonnegative matrices
I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the ...
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On the real and finite field rank of a $0/1$ matrix - II
Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$.
Fix a permutation ...
1
vote
1
answer
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Are certain normal matrices circulant? (Part 2)
Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $
A^TA=\begin{pmatrix}
a & b \\
b & \ddots
\end{pmatrix}$, for $b>0$.
As a user observed in the solution of Part 1 ...