6.2 Diagonalization of Matrices

Given an n×n matrix A, we may ask how many linearly independent eigenvectors the matrix A has. In Section 6.1, we saw several examples (with n=2 and n=3) in which the n×n matrix A has n linearly independent eigenvectors—the largest possible number. By contrast, in Example 5 of Section 6.1, we saw that the 2×2 matrix

A=[2302]

has the single eigenvalue λ=2 corresponding to the single eigenvector v=[10]T.

Something very nice happens when the n×n matrix A does have n linearly independent eigenvectors. Suppose that the eigenvalues λ1,λ2,,λn (not necessarily distinct) of A correspond to the n linearly independent eigenvectors v1,v2,,vn, respectively. Let

P=[|||v1v2vn|||] (1)

be the n×n matrix having these eigenvectors ...

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