Diagonalization
Summary
diagonalizable: A square matrix A is diagonalizable if A is similar to a diagonal matrix. That is, if for some diagonal matrix D.
The Diagonalization Theorem: is diagonalizable if and only if A has n linearly independent eigenvectors. Moreover, (where D is diagonal) if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are the eigenvalues.
Example: #2, p. 325
Rewrite the equation in the form AP=PD to understand what is going on! This is just the eigenvalue equation in partitioned form:
Theorem 6: An matrix with n distinct eigenvalues is diagonalizable.
Example: #10, p. 326
Theorem 7: Let A be an matrix whose distinct eigenvalues are .
Example: #33, p. 326