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Here is an example which shows that full-rank $C$ is not always sufficient to ensure that $W\cdot C$ has full rank. Both $W$ and $C$ have full rank, but $W\cdot C$ does not. $$W = \begin{pmatrix} 1 & …

Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals. If $p_1 \ne p_2$, then there is some positive inte …

There are zero or infinitely many solutions depending on where the non-zero entries have to be. So there is no general-purpose answer. I don't think your equations are properly stated, as $\boldsymbol …

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 …

Restricting $Y$ to an annulus doesn't seem useful as any bounds are likely to be satisfied also inside the annulus. A bound with $Y$ restricted to a disk, or more generally to a region with bounded di …

(A comment rather than an answer.) Here is a plot of $a_n/n^5$ (red) and $b_n/n^5$ (blue). It might not go far enough to show the asymptotic behaviour, but a possibility is that $a_n$ and $b_n$ are as …

[EXPANDED] PART 1 (also done by Peter) If $x,y$ are coprime and have opposite sign, there is a singular symmetric matrix with 1 on the diagonal and only $x$ and $y$ off the diagonal. Say $x<0,y>0$. Si …

This is an example of generating a random point in a polytope. There is some work on that general problem, but I don't know if it would be practical for your special case. For example: Rubin, Generat …

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 …

Let each $A_i$ be a matrix with all entries 0 except for the $(i,i+1)$ entry which is 1, where $i+1=1$ if $i=n$. The characteristic polynomial of $\sum_j a_j A_j$ is $x^n - \prod_j a_j$. I believe tha …

If there is a rational nonzero solution, there is an integer nonzero solution by multiplying up. At least one of the integers can be assumed odd by dividing out a common power of two. The determinant …

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 …

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 …

$$B=\left[\matrix{1&1&1&1\\1&1&-1&-1\\-1&-1&-1&1\\-1&1&-1&1}\right]$$ and probably many other solutions. I'm also voting to close because you didn't pose a research-level problem. If you have an int …

You are asking for relationships between the maximum independent set and the eigenvalues. If you search with those terms you will find several. For example, Haemers proved that the maximum size of a …