Calculate eigenvalues 1 and 2 and obtain the corresponding

Question

Question 1:

Consider the matrix A given by

(a) calculate eigenvalues λ₁ and λ₂ and obtain the corresponding normalized eigenvectors v^₁ and v^₂ for A. It should be apparent at the outset, from the structure of A, that the eigenvectors may realized as an orthonormal set. Why?

(b) Show by direct calculation that A may be written as

A = U†DU 

where U is a unitary matrix and D is a matrix with nonzero entries only on the main diagonal.

Question 2:

Use the results of 1) to show that

where cosh(x) = (e^x + e^-x)/2 and sinh(x) = (e^x - e^-x)/2.