Set
A = ones(10) + eye(10)
(a) What is the rank of A - I? Why must λ = 1 be an eigenvalue of multiplicity 9? Compute the trace of A using the MATLAB function trace. The remaining eigenvalue λ10 must equal 11. Why? Explain. Compute the eigenvalues of A by setting e = eig(A). Examine the eigenvalues using format long. How many digits of accuracy are there in the computed eigenvalues?
(b) The MATLAB routine for computing eigenvalues is based on the QR algorithm described in Section 6 of Chapter 7. We can also compute the eigenvalues of A by computing the roots of its characteristic polynomial. To determine the coefficients of the characteristic polynomial of A, set p = poly(A). The characteristic polynomial of A should have integer coefficients. Why? Explain. If we set p = round(p), we should end up with the exact coefficients of the characteristic polynomial of A. Compute the roots of p by setting
r = roots(p)
and display the results using format long. How many digits of accuracy are there in the computed results? Which method of computing eigenvalues is more accurate, using the eig function or computing the roots of the characteristic polynomial?