Answer c and d c show that despite being

• Peterliao
• 196
• 100% (1) 1 out of 1 people found this document helpful

This preview shows page 187 - 190 out of 196 pages.

We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 11 / Exercise 12
Differential Equations with Boundary-Value Problems
Zill
Expert Verified
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=.
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 11 / Exercise 12
Differential Equations with Boundary-Value Problems
Zill
Expert Verified
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.