MATH 405Final ExamW. AdamsDecember 20, 2002Show all work. Justify all answers.This is an open book exam.1. (20 points each) LetA="0i-i0#.(a) Show thatAis unitary.(b) Find a diagonal matrixDand a unitary matrixUsuch thatA=UDU*.2. (15 points each) Assume that the minimal polynomial,mA, and characteristic polyno-mial,PAof a matrixAare as given in the three cases below. Describe or write downall possible Jordan canonical (normal) forms forA.(a)PA(t) = (t-3)4(t-2)3andmA(t) = (t-3)3(t-2).(b)PA(t) = (t-2)5andmA(t) = (t-2)3.(c)PA(t) = (t-λ1)e1(t-λ2)e2· · ·(t-λm)emandmA(t) = (t-λ1)e1-1(t-λ2)e2-1· · ·(t-λm)em-1for integerse1≥2, . . . , em≥2.3. (25 points) Find the Jordan canonical form of the matrixA=-409010-408.4. (25 points) LetVbe a vector space of dimension 3 and letTbe a linear operator onV. Assume that we have a basisv1, v2, v3ofVsuch that the matrix ofTwith respectto this basis is the matrixAin the last problem. Show that 3v1+ 2v3is an eigenvectorforT. What is the corresponding eigenvalue?
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