# Tagged Questions

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

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### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ...
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### How much linear algebra can be done with graphs?

Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...
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### Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when they are Hermitian and positive-definite? I am investigating ...
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### Why are tensors a generalization of scalars, vectors, and matrices?

Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear ...
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### Elementary applications of linear algebra over finite fields

I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
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### A slick proof of the Bruhat Decomposition for GL_n(k)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...
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### When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...