1. Problem 1
Consider the matrix $A = \begin{bmatrix} 1 & -1 & 0 \ -1 & 2 & -1 \ 0 & -1 & 1 \end{bmatrix}$, answer the following questions
(a) Find three eigenvalues $\lambda_1, \lambda_2, \lambda_3$ and their corresponding eigenvectors $v_1, v_2, v_3$
(b) calculate the inner product (dot product) of three eigenvectors, $v_1 \cdot v_2$, $v_2 \cdot v_3$, $v_1 \cdot v_3$
Are these three eigenvectors orthogonal with each other?