DiagonalizationDiagonal matrices are very easy to work wit h.Example 1. For instance, it is easy to compute the ir powers.IfA =2 0 00 3 00 0 4, then A100=Example 2. If A =6 −12 3, then A100= ?Solution.The key idea of the previous example was to work with respect to a basis given by theeigenvectors.• Put the eig e n vectors x1,, xnas columns into a matrix P .Axi= λxiA| |x1xn| |=| |λ1x1λnxn| |• In summary: AP = PDArmin [email protected] that A is n × n and has independent eigenvectors v1,, vn.ThenA can be diagonalized as A = PDP−1.• the columns of P are• the diagonal matrix D hasSuch a diagonalization is possible if and only ifA has enough eigenvectors.Example 3. Diagonalize the following matrix, if pos sib le.A =2 0 01 2 1−1 0 1Solution.Armin [email protected] 4. Matrices A and B are similar if the re i s an invertible matrix P such thatA = PBP−1Note that, in that case, B = P−1A P . So the definition works both ways.Example 5. So, another way to say that a matrix A can be diagonalized as A = PDP−1is that A is similar to a diagonal matrix.In that case, what is An?Solution.Theorem 6. Similar matrices have the same characteristic polynomial (and hence thesame eigenvalues).Proof. Suppose that A = PBP−1. Practice problemsProblem 1. Find, if possible, the diagonalization of A =0 −2−4 2.Problem 2. Find, if possible, the diagonalization of A =1 2 10 −5 01 8 1.Armin
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