Answer c and d c show that despite being

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Differential Equations with Boundary-Value Problems
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Chapter 11 / Exercise 12
Differential Equations with Boundary-Value Problems
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Answer:c=andd=.(c) Show that despite being diagonalizable (as an operator onC2)Tis not normal.Answer:TT*=6==T*T.(d) Explain briefly why the result of part (c) does not contradict Theorem27.1.3.(5) LetTbe the operator onC3whose matrix representation is168-i5-2i2 + 4i-5 + 2i8-i-4 + 2i-2-4i-4 + 2i14 + 2i.(a) Find the characteristic polynomial and minimal polynomial forT.Answer:cT(λ) =.mT(λ) =.(b) The eigenspaceM1associated with the real eigenvalueλ1is the span of (1 ,,).(c) The eigenspaceM2associated with the complex eigenvalueλ2with negative imaginarypart is the span of ( 1 ,,).(d) The eigenspaceM3associated with the remaining eigenvalueλ3is the span of ( 1 ,,).(e) Find the (matrix representations of the) orthogonal projectionsE1,E2, andE3ontothe eigenspacesM1,M2, andM3, respectively.Answer:E1=1m1-bbcba-c-bc-cd;E2=1n1be-baeeee;E3=1p1-b-bbaabaawherea=,b=,c=,d=,e=,m=,n=, andp=.(f) WriteTas a linear combination of the projections found in (e).Answer: [T] =E1+E2+E3.(g) Find a unitary matrixUwhich diagonalizesT. What is the associated diagonal formΛ ofT?Answer:U=abcabacdbc-dbdc-bdbcebdcand Λ =λ000μ000νwherea=,b=,c=,d=,e=,λ=,μ=, andν=.(h) The operatorTis normal becauseTT*=T*T=16a-bc2bcbca-2b-2bc-2bdwherea=,b=,c=, andd=.
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Differential Equations with Boundary-Value Problems
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Chapter 11 / Exercise 12
Differential Equations with Boundary-Value Problems
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18027. SPECTRAL THEOREM FOR COMPLEX INNER PRODUCT SPACES(6) LetTbe the operator onC3whose matrix representation is135 + 2i2-i2-i2-i5-i2 + 2i2-i2 + 2i5-i.(a) Find the characteristic polynomial and minimal polynomial forT.Answer:cT(λ) =.mT(λ) =.(b) The eigenspaceM1associated with the real eigenvalueλ1is the span of (1 ,,).(c) The eigenspaceM2associated with the complex eigenvalueλ2with negative imaginarypart is the span of (,,-1).(d) The eigenspaceM3associated with the remaining eigenvalueλ3is the span of (,-1 ,).(e) Find the (matrix representations of the) orthogonal projectionsE1,E2, andE3ontothe eigenspacesM1,M2, andM3, respectively.Answer:E1=1maaaaaaaaa;E2=1nbbbbc-cb-cc;E3=16d-e-e-eaa-eaawherea=,b=,c=,d=,e=,m=, andn=.(f) WriteTas a linear combination of the projections found in (e).Answer: [T] =E1+E2+E3.(g) Find an orthogonal matrixQ(that is, a matrix such thatQt=Q-1) which diagonal-izesT. What is the associated diagonal form Λ ofT?Answer:Q=ab0cbcabac-abcab-ac-abcand Λ =λ000μ000νwherea=,b=,c=,λ=,μ=, andν=.
27.3. PROBLEMS18127.3. Problems(1) LetNbe a normal operator on a finite dimensional complex inner product space V. ShowthatkNxk=kN*xkfor allxV.

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