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 ith eigenvector has 1 for its ith 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 ith eigenvector is the ith diagonal element of the matrix. Accordingly, the diagonal elements of the matrix are its eigenvalues.

 

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