8.6: The Eigenvalue Problem- Exercises
Argue as in Proposition 1 in the discussion of the partial fraction expansion of the transfer function that if \(j \ne k\) then \(D_{j}P_{k} = P_{j}D_{k} = 0\).
Argue from the equation from the discussion of the Spectral Representation that \(D_{j}P_{j} = P_{j}D_{j} = D_{j}\).
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 \(m_{1} = m_{2} = 2\). 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
\[B = \begin{pmatrix} {2}&{8}&{6}&{4}\\ {4}&{2}&{8}&{6}\\ {6}&{4}&{2}&{8}\\ {8}&{6}&{4}&{2} \end{pmatrix} \nonumber\]
Carefully label all eigenvalues, eigenprojections and eigenvectors.