## 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_