Abstract
Thus, if A is diagonalizable, then A has n linearly independent eigenvectors. Conversely, if A has n linearly independent eigenvectors u1, . . . , un and if λ1, . . . , λn are the associated eigenvalues, possibly repeated, then taking U with columns u1, . . . , un, and D as the diagonal matrix with diagonal entries λ1, . . . , λn, we have AU = UD, i.e., A = UDU−1. In fact, D is the matrix representation of A with respect to the basis consisting of columns of U . From the above discussion we have the following characterization for diagonalizability of A:
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