5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION
DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields A 7 2 = 4 1 A = PDP 1 5 0 D = 0 3 P 2 1 1 = 1 1 Slide 5.3-2
DIAGONALIZATION The, by associativity of matrix multiplicatio, A = ( PDP )( PDP ) = PD( P P) DP = PDDP 123 2 1 1 1 1 1 Agai, 2 1 1 5 0 2 1 2 1 = PD P = 2 1 2 0 3 1 1 A = ( PDP ) A = ( PD { P ) P D P = PDD P = PD P 3 1 2 1 2 1 2 1 3 1 I I Slide 5.3-3
DIAGONALIZATION k 1 k 1 1 5 0 2 1 k k 1 A = PD P = k 1 2 0 3 1 1 I geeral, for, k k k k 2 5 3 5 3 = k k k k 2 3 2 5 2 3 5 A square matrix A is said to be diagoalizable if A is similar to a diagoal matrix, that is, if A = PDP 1 for some ivertible matrix P ad some diagoal, matrix D. Slide 5.3-4
THE DIAGONALIZATION THEOREM Theorem 5: A matrix A is diagoalizable if ad oly if A has liearly idepedet eigevectors. A = PDP 1 I fact,, with D a diagoal matrix, if ad oly if the colums of P ad liearly idepedet eigevectors of A. I this case, the diagoal etries of D are eigevalues of A that correspod, respectively, to the eigevectors i P. I other words, A is diagoalizable if ad oly if there are eough eigevectors to form a basis of. We call such a basis a eigevector basis of. Slide 5.3-5
THE DIAGONALIZATION THEOREM Proof: First, observe that if P is ay matrix with colums v 1,, v, ad if D is ay diagoal matrix with diagoal etries 1,,, the while [ v v L v ] [ v v L v ] AP = A = A A A PD 1 2 1 2 0 L 0 1 0 L 0 M M M 0 0 L 2 = P = ----(1) [ v v L v ] 1 1 2 2 ----(2) Slide 5.3-6
THE DIAGONALIZATION THEOREM Now suppose A is diagoalizable ad right-multiplyig this relatio by P, we have AP = PD. I this case, equatios (1) ad (2) imply that Equatig colums, we fid that A = PDP 1 Sice P is ivertible, its colums v 1,, v must be liearly idepedet.. The [ Av Av L A v ] = [ v v L v ] 1 2 1 1 2 2 Av = v, Av = v, K, Av = v 1 1 1 2 2 2 ----(3) ----(4) Slide 5.3-7
THE DIAGONALIZATION THEOREM Also, sice these colums are ozero, the equatios i (4) show that 1,, are eigevalues ad v 1,, v are correspodig eigevectors. This argumet proves the oly if parts of the first ad secod statemets, alog with the third statemet, of the theorem. Fially, give ay eigevectors v 1,, v, use them to costruct the colums of P ad use correspodig eigevalues 1,, to costruct D. Slide 5.3-8
THE DIAGONALIZATION THEOREM By equatio (1)(3),. This is true without ay coditio o the eigevectors. If, i fact, the eigevectors are liearly idepedet, the P is ivertible (by the Ivertible Matrix Theorem), ad implies that. AP = AP PD = PD A = PDP 1 Slide 5.3-9
DIAGONALIZING MATRICES Example 2: Diagoalize the followig matrix, if possible. 1 3 3 A = 3 5 3 3 3 1 That is, fid a ivertible matrix P ad a diagoal matrix D such that A = PDP 1. Solutio: There are four steps to implemet the descriptio i Theorem 5. Step 1. Fid the eigevalues of A. Here, the characteristic equatio turs out to ivolve a cubic polyomial that ca be factored: Slide 5.3-10
DIAGONALIZING MATRICES 3 2 0 = det( A I) = 3 + 4 = ( 1)( + 2) 2 The eigevalues are ad. Step 2. Fid three liearly idepedet eigevectors of A. Three vectors are eeded because A is a 3 3 matrix. This is a critical step. = 1 = 2 If it fails, the Theorem 5 says that A caot be diagoalized. Slide 5.3-11
DIAGONALIZING MATRICES Basis for Basis for 1 = 1: v = 1 1 1 1 2 : v 1 = = 2 0 ad 1 v = 0 3 1 You ca check that {v 1, v 2, v 3 } is a liearly idepedet set. Slide 5.3-12
DIAGONALIZING MATRICES Step 3. Costruct P from the vectors i step 2. The order of the vectors is uimportat. Usig the order chose i step 2, form P 1 1 1 = [ v v v ] = 1 1 0 1 2 3 1 0 1 Step 4. Costruct D from the correspodig eigevalues. I this step, it is essetial that the order of the eigevalues matches the order chose for the colums of P. Slide 5.3-13
DIAGONALIZING MATRICES Use the eigevalue = 2 twice, oce for each of the eigevectors correspodig to = 2 : 1 0 0 D = 0 2 0 0 0 2 1 To avoid computig, simply verify that. P AD = PD Compute 1 3 3 1 1 1 1 2 2 AP = 3 5 3 1 1 0 = 1 2 0 3 3 1 1 0 1 1 0 2 Slide 5.3-14
DIAGONALIZING MATRICES PD 1 1 1 1 0 0 1 2 2 = 1 1 0 0 2 0 = 1 2 0 1 0 1 0 0 2 1 0 2 Theorem 6: A matrix with distict eigevalues is diagoalizable. Proof: Let v 1,, v be eigevectors correspodig to the distict eigevalues of a matrix A. The {v 1,, v } is liearly idepedet, by Theorem 2 i Sectio 5.1. Hece A is diagoalizable, by Theorem 5. Slide 5.3-15
MATRICES WHOSE EIGENVALUES ARE NOT DISTINCT It is ot ecessary for a matrix to have distict eigevalues i order to be diagoalizable. Theorem 6 provides a sufficiet coditio for a matrix to be diagoalizable. If a matrix A has distict eigevalues, with correspodig eigevectors v 1,, v, ad if P = [ v L v ], the P is automatically ivertible 1 2 because its colums are liearly idepedet, by Theorem 2. Slide 5.3-16
MATRICES WHOSE EIGENVALUES ARE NOT DISTINCT Whe A is diagoalizable but has fewer tha distict eigevalues, it is still possible to build P i a way that makes P automatically ivertible, as the ext theorem shows. Theorem 7: Let A be a matrix whose distict eigevalues are 1,, p. 1 k p a. For, the dimesio of the eigespace for k is less tha or equal to the multiplicity of the eigevalue k. Slide 5.3-17
MATRICES WHOSE EIGENVALUES ARE NOT DISTINCT b. The matrix A is diagoalizable if ad oly if the sum of the dimesios of the eigespaces equals, ad this happes if ad oly if (i) the characteristic polyomial factors completely ito liear factors ad (ii) the dimesio of the eigespace for each k equals the multiplicity of k. c. If A is diagoalizable ad Β k is a basis for the eigespace correspodig to Β k for each k, the the total collectio of vectors i the sets Β 1,, Β p forms a eigevector basis for. Slide 5.3-18