ST259 Lecture Notes - If And Only If
Document Summary
Defatalinearoperatortom a finite dimensional vector space v is called diagonalizable if there is an ordered basis b for v such that it is a diagonal matrix a square matrix a iscalled diagonalizable if la is diagonalizable. D is theeigenvaluecorrespondingto v for 1ej 2 n. Todiagonalize a matrix or a linearoperator is to find a basisof eigenvectors and thecorresponding eigenvalues. The5. 2let acmmm f then a scalar x is an eigenvalue of a iff detca 1in. Defatet acmmm f thepolynomialofct detca t i n is calledthecharacteristic polynomial. The 5. 2 statesthatthe eigenvalues of a matrix arethe zeros ofits characteristic polynomial. Defaket t be a linear operator on an n dimensional vectorspace v with ordered basis. B wedefinethecharacteristic polynomial f t of t tobethecharacteristic polynomial of. Th 5 et aemmm f a thecharacteristic polynomial of a is a polynomial ofdegree n with leadingcoefficientcdm.
