1.) Proof that the following statements are equivalent 1. is an eigenvalue of for some with .
2. There exists a vector with and .
3. , where is the smallest singular value of .
4. .2.) Let be a simple eigenvalue of a diagonalizable matrix with right and left eigenvectors and , and let be the corresponding eigenvalue of the matrix . Show that to first order it holds that Hint: Consider the perturbed eigenvalue equation .
3.) Proof that for and with . Give an example for which .
Programming Exercise
1.) In the submodule scipy.linalg.blas you find a lowlevel interface to the BLAS. Use this interface to make accurate timing measurements of BLAS3 matrix matrix products for real double precision matrices of growing dimension. Plot the time against the growing dimension using a logarithmic scale in the time. (the plot command semilogy is helpful here). Also, plot the function against the dimension and compare the two. What do you observe? [5 marks] z A + δA δA ?δA?2 ≤ ? u ∈ C m ?(A - zI)u?2 ≤ ? ?u?2 = 1 σn(zI - A) ≤ ? σn A ?(zI - A) -1?2 ≥ ? -1 λ A x y λ + δλ A + δA δλ = . y δAx H y x H (A + δA)(x + δx) = (λ + δλ)(x + δx