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Problem of the week

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.