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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. Also, ifx v , then(Tx,v) = (x,T v) = (x,v) = isT-invariant. Then the restriction ofTtov is a normal operator on ann 1 dimen-sional space.
Christian Parkinson UCLA Basic Exam Solutions: Linear Algebra 1 Problem F02.10. Let Tbe a linear operator on a nite dimensional complex inner prod-uct space V such that T T= TT . Show that there is an orthonormal basis of V consisting of eigenvectors of B. Solution. When Tsatis es T T= TT , we call Tnormal.
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