If M is a linear operator on bbbR^3 with unique and real eigenvalues lambda_1<lambda_2<lambda_3, such that EE x in bbbR^3 \\{0}, satisfying the condition lim_(n to oo)norm(M^n x)=0. What are the possible values of lambda_1?

Parker Pitts 2022-10-01 Answered
If M is a linear operator on R 3 with unique and real eigenvalues λ 1 < λ 2 < λ 3 , such that x R 3 { 0 }, satisfying the condition lim n | | M n x | | = 0. What are the possible values of λ 1 ?
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Answers (1)

Lamar Esparza
Answered 2022-10-02 Author has 8 answers
That's not as straightforward of an answer as it seems.
If x is a linear multiple of ζ 1 , then we can affirm that | λ 1 | < 1
Otherwise, we have x = c 1 ζ 1 + c 2 ζ 2 + c 3 ζ 3 and that means M n x = c 1 λ 1 n ζ 2 + c 2 λ 2 n ζ 2 + c 3 λ 3 n ζ 3 , implying all λ i should have magnitude less than 1 to satisfy that property for any arbitrary vector x.
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