# hw3-solutions - MATH 53H, PROBLEM SET #3 SOLUTIONS 6.12....

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MATH 53H, PROBLEM SET #3 — SOLUTIONS6.12. ComputeeA(etA?), whereAis given.(b)A=parenleftbigg2-112parenrightbigg. We havepA(λ) = (λ-2)2+ 1, which has rootsλ= 2±iand correspondingeigenvectorsz±= (±i,1)T. ThusA=PDP-1whereP=parenleftbiggi-i11parenrightbigg,D=parenleftbigg2 +i002-iparenrightbigg,P-1=12parenleftbigg-i1i1parenrightbigg,andetA=PetDP-1=e2t2parenleftbiggi-i11parenrightbiggparenleftbiggeit00e-itparenrightbiggparenleftbigg-i1i1parenrightbigg=e2t2parenleftbiggeit+e-iti(eit-e-it)-i(eit-e-it)eit+e-itparenrightbigg=e2tparenleftbiggcost-sintsintcostparenrightbigg,using the standard formulascosθ= Ree=e+e-2andsinθ= Ime=e-e-2i=i2(e--e).Alternatively, letB= 2IandC=parenleftbigg0-110parenrightbigg, so thatA=B+CandBC=CB. ThenC2=-IimpliesetC=summationdisplayj=0(tC)jj!=summationdisplayj=0(-t2)j(2j)!I+summationdisplayj=0(-1)jt2j+1(2j+ 1)!C=parenleftbiggcost-sintsintcostparenrightbiggandetA=etBetC=e2tetCas above.(c)A=parenleftbigg2-102parenrightbigg. The only eigenvalue ofAis 2, soL=parenleftbigg2002parenrightbigg,N=parenleftbigg0-100parenrightbigg,andetA=parenleftbigge2t-

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