Let L be a Leslie matrix with a unique positive eigenvalue 1 Show that if is any other (real or complex) eigenvalue of L, then 1 Write r(cos i sin ) and substitute it into the equation g() 1, as in part (b) of Exercise 23 Use De Moivres Theorem and then take the real part of both sides The Triangle...

1 r 1 cos 0 i sin 0 r 1 since it is positive Let r cos i sin be some other ...

Let L be a Leslie matrix with a unique positive eigenvalue 1 . Show that if

Question:

Let L be a Leslie matrix with a unique positive eigenvalue λ₁. Show that if λ is any other (real or complex) eigenvalue of L, then |λ| ≤ λ₁. [Write λ = r(cosθ + i sin θ) and substitute it into the equation g(λ) = 1, as in part (b) of Exercise 23. Use De Moivre’s Theorem and then take the real part of both sides. The Triangle Inequality should prove useful.]

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