The Eigenvalue Problem: Exercises 2.2 2002/02/04 2002/07/19 00:00:00.002 GMT-5 Steven Cox cox@rice.edu Charlet Reedstrom charlet@rice.edu Steven Cox cox@rice.edu Eigenvalue eigenvector (Blank Abstract)

Exercises Argue as in Proposition 1 in the discussion of the partial fraction expansion of the transfer function that if j k then Dj Pk Pj Dk 0 . Argue from this equation from the discussion of the Spectral Representation that Dj Pj Pj Dj Dj . The two previous exercises come in very handy when computing powers of matrices. For example, suppose B is 4-by-4, that h 2 and m1 m2 2 . Use the spectral representation of B together with the first two exercises to arrive at simple formulas for B 2 and B 3 . Compute the spectral representation of the circulant matrix B 2 8 6 4 4 2 8 6 6 4 2 8 8 6 4 2 Carefully label all eigenvalues, eigenprojections and eigenvectors.