Show that the eigenvalues of the matrix
are 2, 1 and -1, and find the corresponding eigenvectors. Write down the modal matrix M and spectral matrix Λ of A, and verify that MΛ = AM.
Deduce that the system of difference equations
x(k + 1) = Ax(k)
where x(k) = [x1(k) x2(k) x3(k)]T , has a solution
x(k) = My(k)
where y(k) = Λk y(0). Find this solution, given x(0) = [1 0 0]T .