Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same). 1. Does there exist an isomorphi …

Recall that a chain complex is a (finite) diagram of the form $$ V = \{ \dots \to V_3 \overset{d_3}\to V_2 \overset{d_2}\to V_1 \overset{d_1}\to V_0 \to 0 \} $$ where the $V_n$ are …

This is something I am stuck on (it might well be trivial- in which case this is an embarassing question): Let V be a dimension r vector space over Fp, the field with p prime eleme …

I am asking for a way to compute the rank of the `join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just dim(A_1+A_2)= dim(A_ …

How to compute the dimension of A1+A2+...+An, when they are not direct sum?

In complex vector space, the characteristic polynomial of a operator has n roots. So the operator has n eigenvalues. Obviously, the operator has a upper triangular matrix. If n roo …

If Q is a real homogeneous quartic on $R^N$, $Q(x) = \sum_{1 <= i,j,k,l <= N} Q_{ijkl} x_i x_j x_k x_l$ what is the condition on the (totally symmetric) coefficients $ …

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) pe …

Do you know any good reference on multilinear algebra?

We consider $n\times n$ complex matrices. If $\operatorname{rank}(A)=\operatorname{rank}(B)$, does it imply that there exist $A^-$, $B^-$ such that $A^-+B^-=A^-(A+B)B^-$ holds. Her …

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in ver …

Let $ \lambda_1 \ge \lambda_2 \ge \dots \lambda_{2n} $ be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any w …

We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is wel …

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the mor …

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes: Given a finite (nonempty) set of real numbers, …