Exercise Solution 2.8

Consider an n-dimensional diagonal matrix. We identify n distinct eigenvectors as follows. The first has 1 for its first component and 0 for the rest of its components. The second has 1 for its second component and 0 for the rest of its components. In general, the i^th eigenvector has 1 for its i^th component and 0 for the rest of its components. The eigenvectors are linearly independent, so they are distinct. An n-dimensional matrix cannot have more than n distinct eigenvectors, so these are all the eigenvectors of our matrix. Clearly, the eigenvalue corresponding to the i^th eigenvector is the i^th diagonal element of the matrix. Accordingly, the diagonal elements of the matrix are its eigenvalues.

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