Question

2. in problem 5.25 b)

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Review Exercises 309 (c) Why is eA unitary? (d) Why is eKt unitary? 5.21 (a) Find a nonzero matrix N such that N3 0. (b) If Nx = Ax, show that λ must be zero. (c) Prove that N (called a "nilpotent" matrix) cannot be symmetric 5.22 Suppose the first row of A is 7, 6 and its eigenvalues are i, -i. Find A. 5.23 If the vectors xi and x2 are in the columns of S, what are the eigenvalues and eigenvectors of B-S101 S-1 and S-1? 5.24 If A1, what are the eigenvalues of A? If A is a real n by n matrix show that n must be even, and give an example. 5.25 (a) For which numbers c and d does A have real eigenvalues and orthogonal eigenvectors? 1 2 0 A=12 d c tb) For which c and d can we find three orthonormal vectors that are combinations of the columns (don't do it!)? 5.26 A variation on the Fourier matrix is the "sine matrix": | sin θ V2 sin 2θ sin 3θ 4. sin 3θ sin60 sin 9θ Verify that ST S-1. (The columns are the eigenvectors of the tridiagonal -1,2, I matrix.) 5.27 What is the limit as k → oo (the Markov steady state) of 5.28 If M is the diagonal matrix with entries d, d2, d3, what is M-AM? What are it eigenvalues in the following case?

Answer 1

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