26.2 A be the 32*32 matrix with -1 on the main diagonal, 1 on the first and second superdiagonals, and 0 elsewhere.

a) using an svd algorithm build into MAtlab or another s

m an egenalue A)-1 Ain iv) is known as the resolvent of A at z: if of A, we use the convention lleI-A)- is 1(21-A)-illa oc. In (iii), 0m tes the smallest singular value. that conditions ()-(iv) are equivalent. Let A be the 32 x 32 matrix with -1 on the main diagonal, 1 on the ist and second superdiagonals, and 0 elsewhere. a Using an SVD algorithm built into MATLAB or another software system tngether with contour plotting software, generate a plot of the boundaries of the 2-norm e-pseudospectra of A for 10-1,102, 10- b) Produce a semilogy plot of let 4|12 against t for 0St 50. What is the nitial growth rate of the curve before the eventual decay sets in? Can you relate this to your plot of pseudospectra? (Compare Exercise 24.3.) ne of the best known results of eigenvalue perturbation theory is the e arbitrary. Then every eigenvalue of A+d.A lies im at V)6Alla and et heorem. Suppose A E cmxm is diagonalizable with A VA let 6A mXm b bust one of m ethe m circular disks in the complez plane of radius x( centered at the (Co nualues of A, where is the 2-norm condition number. onbare Ene eige