ST259 Lecture : 5.2 — Diagonalizability
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Tha 5. 5let t be a linearoperator on a vectorspace v and let eigenvalues oft if v vz. Viciti ek then vi va ve is linearlyindependent uh are eigenvectors of t suchthat xicorresponds to. Corollar et t be a linearoperator on an n dimensional vector space v if t has n distinct eigenvalues then t is diagonalizable. Defata polynomial f t in p f splits over f ifthere are scalars c a an in f such that f t oft a ct az ft an. Defiiet 1 be an eigenvalueofa linearoperator or matrix with characteristic polynomial htt the algebraic multiplicity of x isthe largest positive integer k forwhich t d2is afactor of f t. Defket tbe a linearoperator on a vector space v and let 1 be an eigenvalue of t define. Ex kev t x xx n t xlv theset ex is calledtheeigenspace of tcorresponding totheeigenvalue x analogously we definetheeigenspaceof a square matrix a to bethe eigenspace of la.
