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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.

Christian Parkinson UCLA Basic Exam Solutions: Linear Algebra 4 since T(v i) = iv i;i= 1;:::;n: Problem S03.9. Let A2M 3(R) satisfy det(A) = 1 and AtA= I= AAt where Iis the identity matrix. Prove that the characteristic polynomial of Ahas 1 as a root. Solution. Clearly the characteristic polynomial of Ahas a real root since it has odd order.

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