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# The Eigenvalue Problem: Exercises

Module by: Steven J. Cox. E-mail the author

Summary: (Blank Abstract)

## Exercises

1. Argue as in Proposition 1 in the discussion of the partial fraction expansion of the transfer function that if jk j k then Dj Pk = Pj Dk =0 Dj Pk Pj Dk 0 .
2. Argue from this equation from the discussion of the Spectral Representation that Dj Pj = Pj Dj = Dj Dj Pj Pj Dj Dj .
3. The two previous exercises come in very handy when computing powers of matrices. For example, suppose BB is 4-by-4, that h=2 h 2 and m1 = m2 =2 m1 m2 2 . Use the spectral representation of BB together with the first two exercises to arrive at simple formulas for B2 B 2 and B3 B 3 .
4. Compute the spectral representation of the circulant matrix B=( 2864 4286 6428 8642 ) B 2 8 6 4 4 2 8 6 6 4 2 8 8 6 4 2 Carefully label all eigenvalues, eigenprojections and eigenvectors.

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