Transcription of Christian Parkinson UCLA Basic Exam Solutions: Linear ...
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Christian Parkinson UCLA Basic Exam Solutions: Linear ...
Christian ParkinsonUCLA Basic Exam Solutions: Linear Algebra1 Problem a Linear operator on a finite dimensional complex inner prod-uct spaceVsuch thatT T=TT . Show that there is an orthonormal basis ofVconsistingof eigenvectors T=TT , we prove this by induction on the dimension of the space thatToperates on. IfTisoperating on a 1-dimensional space, the claim is the claim holds for any normalToperating on ann 1 dimensional space(n 2). By the fundamental theorem of algebra, the characteristic polynomial ofT has aroot which is an eigenvalue ofT . Letvbe the corresponding non-zero eigenvector (wlog||v||= 1). Thenv ={x V: (x,v) = 0}has dimensionn 1.
jjSxjj2 = (Sx;Sx) = (x;SSx) = (x;SSx) = (Sx;Sx) = jjSxjj2: Then since vis an eigenvector of T , we have (T I)v= 0:Then 0 = jj(T I)vjj2 = jj(T I) vjj2 = (T I)v 2: Thus Tv= vso vis also an eigenvector of T. Thus fv;v 2;:::;v ngis a basis of V consisting of eigenvectors of T. This completes the induction and the proof. Problem W02.8.
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