# Tagged Questions

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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) ...
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### When is a matrix power nonnegative

The following question came up today during a discussion: Suppose $A$ is an $n \times n$ real matrix. Is there some way to tell whether there exists an integer $q > 0$ such that $A^q$ is ...
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### Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite. My question is: Is there a ...
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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 ...
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### When is a matrix similar to a non-negative matrix?

Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...
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### Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
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### A conjecture of Blakley and Dixon about odd powers of positive matrices

In a 1966 paper Blakley and Dixon conjecture the following. Let $S$ be a symmetric matrix with nonnegative entries and let $u$ be a unit vector with nonnegative entries. For integers $k\ge j$ both odd,...
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### Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries

This is a more carefully worded version of this question, here tailored to professional mathematicians. Consider a matrix ${\bf A}\in{\bf M}_{n\times n}({\mathbb R})$ with possibly positive, negative ...
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### A matrix inequality involving the Hilbert-Schmidt norm

This question comes from a problem in PDEs on which I'm currently working. Let $a$ be a $3\times 3$ matrix, real symmetric and positive definite. Denote with $\|a\|^2 _ 2=\sum a_{ij}^2$ the square of ...
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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 ...
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### numerical range of a column-zero-sum matrix

I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
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### Non symmetric matrices with real eigenvalues

Consider the following block matrix $A=\pmatrix{A_1 & A_2\cr kA_2^\top & A_3}$ where $A_1$ is a symmetric matrix, $A_3$ is diagonal matrix and all entries of $A$ are real and non-negative. ...
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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 ...
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### Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question: Bounding the absolute sum of entries of the ...
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### Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications? For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature. In some of ...
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### A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $A-B$ is semidefinite (i.e.$A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...
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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 ...
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### nonnegative solution of nonhomogeneous under-determined linear system of equations

For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such ...
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### Showing positive stability of a matrix constructed from a positive matrix

A is a positive nonsingular matrix. Let $s>\rho(A)$. We want to show that $B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right)$ is a positive stable matrix, i.e., all eigenvalues of this matrix ...
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### A Perron-Frobenius problem

Let $A$ be an irreducible nonnegative matrix with column sums equal to 1. Let $b\in R^n$ have components summing to 0, and let $u$ be the solution of $u=Au+b$ with components summing to 1 (unique ...
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### Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form: \begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log \...
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### Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$. Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
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### Bounding the Schur's complement of similiar matrices

Assume the following: • $L\leq K$ . • $\Gamma\in M_{K,L}$ is a $L$ rank ${ 0,1}$ matrix, without identical rows or the zeros row. • $N\in M_{K,K}$ is a diagonal matrix, whose diagonal is a ...
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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 ...
Let $A$ and $B$ be symmetric non-negative matrices. If $A\geq B$ (i.e., $A-B$ is a nonnegative matrix), can we say that $\lambda_i(A) \geq \lambda_i(B)$ for all $i$, where $\lambda_i$ denotes the i--...
Let $\mathcal{N}$ be the space of all $n \times n$ matrices that are similar to some nonnegative matrix with zero diagonal and let $f: \mathcal{N} \to \mathbb{R}$ be a continuously differentiable ...