Transcription of EIGENVALUES AND EIGENVECTORS - NUMBER THEORY
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EIGENVALUES AND EIGENVECTORS - NUMBER THEORY
Chapter 6 EIGENVALUES MotivationWe motivate the chapter on EIGENVALUES by discussing the equationax2+ 2hxy+by2=c,where not all ofa, h, bare zero. The expressionax2+ 2hxy+by2is calledaquadratic forminxandyand we have the identityax2+ 2hxy+by2= x y a hh b xy =XtAX,whereX= xy andA= a hh b .Ais called the matrix of the now rotate thex, yaxes anticlockwise through radians to newx1, y1axes. The equations describing the rotation of axes are derived asfollows:LetPhave coordinates (x, y) relative to thex, yaxes and coordinates(x1, y1) relative to thex1, y1axes. Then referring to Figure :115116 CHAPTER 6. EIGENVALUES AND EIGENVECTORS - 6? @@@@@@@@I @@@@@@@@R @@@xyx1y1 PQRO Figure : Rotating the ( + )=OP(cos cos sin sin )= (OPcos ) cos (OPsin ) sin =ORcos PRsin =x1cos y1sin .Similarlyy=x1sin +y1cos.
plex column vector satisfying AX = λX, we call X an eigenvector of A, while λ is called an eigenvalue of A. We also say that X is an eigenvector corresponding to the eigenvalue λ. So in the above example P1 and P2 are eigenvectors corresponding to λ1 and λ2, respectively. We shall give an algorithm which starts from the eigenvalues of A ...
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