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

11 votes
3 answers
589 views

Non-singular matrix with restricted entries

Given a set $S$ of integers with $1 \not\in S$, let us consider the set $\mathcal{M}$ of all the symmetric matrices $M$, such that: All the diagonal entries of $M$ are equal to $1$. All the off-diag …
26 votes
1 answer
5k views

Generalization of Cauchy's eigenvalue interlacing theorem?

Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace. …
6 votes
0 answers
381 views

Is there an easy way to compute the maximum isotropic subspace over finite fields?

Given a quadratic form (or a symmetric $n \times n$ matrix $A$), an isotropic subspace is a subspace $U$ such that $$U^t A U=0,$$ If I am not mistaken, when the matrix is over reals, the maximum dime …