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Eigenvalues, Eigenvectors, and Diagonalization

Eigenvalues, Eigenvectors, and Diagonal-izationMath 240 EigenvaluesandEigenvectorsDiagonalizatio nEigenvalues, eigenvectors , and DiagonalizationMath 240 Calculus IIIS ummer 2013, Session IIWednesday, July 24, 2013 Eigenvalues, Eigenvectors, and Diagonal-izationMath 240 EigenvaluesandEigenvectorsDiagonalizatio nAgenda1. Eigenvalues and Eigenvectors2. DiagonalizationEigenvalues, eigenvectors , and Diagonal-izationMath 240 EigenvaluesandEigenvectorsDiagonalizatio nIntroductionNext week, we will apply linear algebra to solving that is particularly easy to solve isy = has the solutiony=ceat, wherecis any real (or complex) in terms of linear transformations,y=ceatisthe solution to the vector equationT(y) =ay,(1)whereT:Ck(I) Ck 1(I)isT(y) =y.

Find all of the eigenvalues and eigenvectors of A= 1 1 0 1 : The characteristic polynomial is ( 1)2, so we have a single eigenvalue = 1 with algebraic multiplicity 2. The matrix A I= 0 1 0 0 has a one-dimensional null space spanned by the vector (1;0). Thus, the geometric multiplicity of this eigenvalue is 1.

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