Let Mn be the n × n tridiagonal matrix whose diagonal entries are all equal to 0 and whose sub- and super-diagonal entries all equal 1.
(a) Find the eigenvalues and eigenvectors of M2 and M3 directly.
(b) Prove that the eigenvalues and eigenvectors of Mn are explicitly given by

for k = 1........n. How do you know that there are no other eigenvalues?

= 2 cos- -(sin "k, san 2k, . sin"+1)