Construct a defective matrix by setting
A = ones(6); A = A - tril(A) - triu(A, 2)
It is easily seen that λ = 0 is the only eigenvalue of A and that its eigenspace is spanned by e1. Verify this by using MATLAB to compute the eigenvalues and eigenvectors of A. Examine the eigenvectors using format long. Are the computed eigenvectors multiples of e1? Now perform a similarity transformation on A. Set
Q = orth(rand(6)); and B = Q′ * A * Q
If the computations had been done in exact arithmetic, the matrix B would be similar to A and hence defective. Use MATLAB to compute the eigenvalues of B and a matrix X consisting of the eigenvectors of B. Determine the rank of X. Is the computed matrix B defective? Because of roundoff error, a more reasonable question to ask is whether the computed matrix B is close to being defective (i.e., are the column vectors of X close to being linearly dependent?). To answer this, use MATLAB to compute rcond(X), the reciprocal of the condition number of X, A value of rcond close to zero indicates that X is nearly rank deficient.