## Problem of the week

### 2019-03 Simple spectrum

Suppose that $$T$$ is an $$N \times N$$ matrix $T = \begin{pmatrix} a_1 & b_1 & 0 & \cdots & 0 \\ b_1 & a_2 & b_2 & \ddots & \vdots \\ 0 & b_2 & a_3 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & b_{N-1} \\ 0 & \cdots & 0 & b_{N-1} & a_N \end{pmatrix}$ with $$b_i > 0$$ for $$i =1, 2, \dots, N-1$$. Prove that $$T$$ has $$N$$ distinct eigenvalues.