# Tagged Questions

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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### Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
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### Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
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### Number of positive eigenvalues of the product of two matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix that has at least one positive eigenvalue with positive real part. Let $B\in\mathbb{R}^{n\times n}$ be a matrix where all its eigenvalues have positive ...
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### Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints

Suppose we have an n by m nonnegative matrix A, where each row sums to 1. I wonder whether there exists an m by n nonnegative matrix X that satisfies the following constraints: each row of X sums to ...
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### Transitivity of $Spin(7)$ in triples of vectors

I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e_1,e_2,e_3$, $v_1,v_2,v_3$ be two triples of mutually ...
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### Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true? $\|A\|_{2}$ denotes ...
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### How do I ensure that my matrix is positive definite? [closed]

I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form $$M=\begin{pmatrix} \Sigma&P'\\ P&\Sigma \end{pmatrix}$$ where $\Sigma$ is a ...
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### Hopfian modules

My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
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### Positive definite - Inverse of sparse symmetric matrix

Consider a matrix $P\in \mathbb{R}^{n\times n}$ such that at most $m<n$ elements of each column are non zero and $P$ is symmetric. I would like to find the sufficient condition(s) such that ...
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### Closed-form solution of a linear programming question

Among all the probability matrices \begin{equation*} P = \left(\begin{array}{cccc} p_{00} & p_{01} & \ldots & p_{0,J-1} \\ p_{10} & p_{11} & \ldots & p_{1,J-1} \\ \vdots & ...
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Consider the $2N\times 2N$ matrix $$A=\begin{pmatrix} a &1 &0&0&0&\ldots&0&1 \\1 &-a&1 & 0 &0 & \ldots & 0&0 \\0 &1&a&1&0 ... 1answer 39 views ### The effect of linear transformation on generic vectors [closed] I have a question about the effect of applying a linear transformation M in \mathbb{R}^{n \times n} to a vector v \in \mathbb{R^n}. I know that if M has p-norm \|M\|_p = \lambda, then by ... 1answer 108 views ### Two minimization problems using singular value decomposition Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition Let q_0, q_1:[0,1]\to \mathbb{R}^n be two maps whose components are ... 1answer 112 views ### Commuting nilpotent matrix collection For every large enough m\in\Bbb N are there c=\alpha m (for some fixed \alpha>0) square matrices A_1,\dots,A_c that commute with each other with nonzero product (\forall ... 1answer 131 views ### Exact eigenvalues of a specific tridiagonal matrix I'm studying the following tri-diagonal matrix$$ X = \begin{pmatrix} 0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\ x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
I have a (directed graded) graph $G=(V,E)$ with two distinguished subsets of $V$: sources and destinations. I am to show that the set of indicator functions of paths starting in a source and ending in ...