Prove Theorem 9.13.
Theorem 9.13
An n × n matrix A is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. In this case, D = S−1 AS, where the columns of S consist of the eigenvectors, and the ith diagonal element of D is the eigenvalue of A that corresponds to the ith column of S.
The pair of matrices S and D is not unique. For example, any reordering of the columns of S and corresponding reordering of the diagonal elements of D will give a distinct pair.
We saw in Theorem 9.3 that the eigenvectors of a matrix that correspond to distinct eigenvalues form a linearly independent set.