Given an matrix A, we may ask how many linearly independent eigenvectors the matrix A has. In Section 6.1, we saw several examples (with and ) in which the matrix A has n linearly independent eigenvectors—the largest possible number. By contrast, in Example 5 of Section 6.1, we saw that the matrix
has the single eigenvalue corresponding to the single eigenvector
Something very nice happens when the matrix A does have n linearly independent eigenvectors. Suppose that the eigenvalues (not necessarily distinct) of A correspond to the n linearly independent eigenvectors , respectively. Let
be the matrix having these eigenvectors ...