lesson28 - MA 511 Session 28 Diagonalization Let A be a n n...

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MA 511, Session 28DiagonalizationLetAbe an×nmatrix. Suppose thatAhasnlinearly independent eigenvectors,v1, . . . , vn, corre-sponding to the eigenvaluesλ1, . . . , λn, respectively.Let us define the eigenvalue matrixΛ and aneigenvector matrixSas follows:Λ =λ10. . .00λ2. . .0···00. . .λn,S= (v1. . .vn).The eigenvalues may be repeated, and may be real orcomplex. Also, since the columns ofSare assumedindependent,Sis invertible and we haveTheorem:A=SΛS-1, that is, ifAhasnlinearlyindependent eigenvectors, thenAis diagonalizable.Proof: SinceAvj=λjvj, 1jn, it follows thatAS=SΛ.
We say thatAhas been diagonalizedbyS. Notall matrices can be diagonalized. The matrixAfromlast session is (trivially) diagonalizable since it is al-ready diagonal, butBandCare not diagonalizable.To better understand this concept, let us in-troduce the concepts of algebraic multiplicityandgeometric multiplicityof an eigenvalueλofA. Theformer is the multiplicity ofλas root of the charac-

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