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Solve $A^T A \boldsymbol{x}=A^T\boldsymbol{e_k}$ first, it is small enough to handle by dense methods. If there is no solution, there is also none for the original. Otherwise get an (affine) basis o …

This is not an answer but a conjecture based on some trials. $$a_{n,k} = \begin{cases} 1, & n=1 \\ 0, & n\gt 1, k\text{ odd} \\ (-1)^{t\binom n2}\di …

Eigenvalues are continuous functions of the matrix entries if this is expressed carefully. Consider $H(c) = cP_1+P_2+\cdots+P_m$. When $c$ is large and negative, the eigenvalues of $H(c)$ are all neg …

Your question is a little unclear, but I take it you want some sort of sampling method that estimates the number of distinct sums. I'll answer a more general question. This method is not as well kno …

You can always do it if there is any eigenvector not summing to 0, and some other times too. First define $Aut(A)$ to be the group of permutation matrices $R$ such that $RAR^T=A$. If $A\boldsymbol{x …

This is an extended comment short of a solution. Define an $n^2\times n^2$ matrix $A_n$ with rows and columns indexed by the positions $(i,j)$. Row $(i,j)$ of the matrix is 0 except for 1 in the col …

In the case of integer $\gamma_i$, the determinant is nonzero (in fact, strictly positive). This is proved in Gantmacher, The Theory of Matrices, Vol 2, p99. Gantmacher calls it a "generalized Vande …

One fuzzy connection: Say matrix $A$ has characteristic polynomial $f(x)$. Then the derivative $f'(x)$ is the sum of the characteristic polynomials of the principal minors $A_{ii}$.

We all know Gershgorin's Circle Theorem, which I will summarise for convenience. Let $A=(a_{ij})$ be an $n\times n$ complex matrix. Define the disks $D_1,\ldots,D_n$ by $$D_i = \Bigl\{ z : |z-a_{ii}| …

Higham, in Functions of Matrices, Theorem 1.12, shows that the Jordan form definition is equivalent to a definition based on Hermite interpolation. That shows that the square root of a matrix $A$ (if …

In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting. Let $\chi(M)$ denote the characteristic pol …

(The first part of this is the same as @Fedor's answer; I just carried out his algorithm.) Each row $i$ can be written as $a_i^{(1)}x_i^{(1)}+a_i^{(2)}x_i^{(2)}$ where $x_i^{(1)},x_i^{(2)}$ are non-ne …

It is undecidable. I'll use this result: Given an finite set of integer matrices, it is undecidable whether there is a product with 0 in the upper-right corner. See for example this article. So tak …

This problem was solved by Ron Read in his PhD thesis (University of London, 1958). Without requirement 2 there is a summation which isn't too horrible. With requirement 2 added as well, Read's soluti …

If $k$ is the greatest common divisor of the cycle lengths of the digraph, then the spectrum is invariant under rotation around the origin by $2\pi/k$. This is an application of the Perron-Frobenius t …