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eigenvalues of tridiagonal matrix

Source: VJIMC 2012 2.2

May 31, 2021
matrixlinear algebra

Problem Statement

Let MM be the (tridiagonal) 10×1010\times10 matrix M=(−130⋯⋯⋯032−10⋮0−12−1⋱⋮⋮0−12⋱0⋮⋮⋱⋱⋱−10⋮0−12−10⋯⋯⋯0−12)M=\begin{pmatrix}-1&3&0&\cdots&\cdots&\cdots&0\\3&2&-1&0&&&\vdots\\0&-1&2&-1&\ddots&&\vdots\\\vdots&0&-1&2&\ddots&0&\vdots\\\vdots&&\ddots&\ddots&\ddots&-1&0\\\vdots&&&0&-1&2&-1\\0&\cdots&\cdots&\cdots&0&-1&2\end{pmatrix}Show that MM has exactly nine positive real eigenvalues (counted with multiplicities).
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