每週問題 February 3, 2014

Without computing the eigenvalues, decide how many are positive, negative, and zero for

$\displaystyle A=\begin{bmatrix} 0&0&\cdots&0&1\\ 0&0&\cdots&0&2\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&0&n-1\\ 1&2&\cdots&n-1&n \end{bmatrix},~~n\ge 2$.

$\displaystyle \lambda_{\min}\le\frac{\mathbf{x}^TA\mathbf{x}}{\mathbf{x}^T\mathbf{x}}\le\lambda_{\max}$