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# 8.6: The Eigenvalue Problem- Exercises

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Exercise $$\PageIndex{1}$$

Argue as in Proposition 1 in the discussion of the partial fraction expansion of the transfer function that if $$j \ne k$$ then $$D_{j}P_{k} = P_{j}D_{k} = 0$$.

Exercise $$\PageIndex{2}$$

Argue from the equation from the discussion of the Spectral Representation that $$D_{j}P_{j} = P_{j}D_{j} = D_{j}$$.

Exercise $$\PageIndex{3}$$

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.