## Eigenvalue Bounds

While grading homeworks today, I came across the following bound:
Theorem 1: If A and B are symmetric $n\times n$ matrices with eigenvalues
$\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_

While grading homeworks today, I came across the following bound:
Theorem 1: If A and B are symmetric $n\times n$ matrices with eigenvalues
$\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_

Today Arun asked me the following question:
"Under what conditions will a set $\{p_1,\ldots,p_n\}$ of polynomials be
quadratically independent, in the sense that $\{p_1^2, p_1p_