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. 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. Then by our inductive hypothesis, there is an orthonormal basis{v2,..,vn}forv consisting of eigenvectors ofT. Then{v,v2,..,vn}is an orthonormal set withnelements and is thus a basis forV.
By the spectral theorem, the eigenspaces corresponding to distinct eigenvalues will be orthogonal. Here all eigenvalues are distinct. Since the rst two eigenvectors span a two dimensional space, any vector orthogonal to both will necessarily be a third eigenvector. Taking the cross product of the two vectors gives a vector which is orthogonal ...
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