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Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equ …

Let's see if I can convince everyone that this problem is NP-complete. First: it is in NP because a permutation $P$ can be guessed and checked in polynomial time. I'll restate the problem: Given a …

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}| …

The smallest eigenvalue can go up or down when an edge is removed. For "down": $G=K_n$ for $n\ge 3$. For "up": Take $K_n$ for $n\ge 1$ and append a new vertex attached to a single vertex of the orig …

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 …

A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}( …

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 …

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 …

It is NP-complete. Consider the case of $n$ vectors of length $n+1$, where the vectors are the rows of a matrix $[I | x]$ where $I$ is an $n\times n$ identity matrix and $x$ is a column vector of eve …

Even the case where $B$ is constant on the diagonal and constant off the diagonal is extremely difficult. For example, it includes the question of for which orders a finite projective plane exists. If …

In general the number of decompositions depends on the structure of the matrix, not just on its size and row sum. This is even so in the case of 0-1 matrices, where the question is equivalent to 1-fa …

For general $A,X~$ this is a very difficult problem, but the condition you give that the rows of $X~$ have sum 2 makes it much easier. Consider each row to be an edge of a graph $G~$ (i.e. the two on …

Let $L_i$ and $R_i$ be the $i$-th rows of the left and right matrices. Then $$\begin{align} R_n &= L_n \\ R_{n-1} &= L_n + L_{n-1} \\ R_{n-2} &= L_n + 2L_{n-1} + L_{n-2} \\ R_{n-3} …

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 …

You have reduced the graph isomorphism problem to a 0-1 programming problem. 0-1 programming problems are NP-hard in general, so the question is whether your particular case is an exception. You haven …