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 .We can combine these transformation equations into the single matrixequation: xy = cos sin sin cos x1y1 ,orX=P Y, whereX= xy , Y= x1y1 andP= cos sin sin cos.
Chapter 6 EIGENVALUES AND EIGENVECTORS 6.1 Motivation We motivate the chapter on eigenvalues by discussing the equation ax2 +2hxy +by2 = c, where not all of a, h, b are zero.
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