Question

(Spectral Mapping Theorem for polynomial function)

Suppose that Aeᵢ = λᵢeᵢ for i = 1,⋯,n. Show that the polynomial matrix
p(A) = kₘAᵐ + kₘ−₁Aᵐ−¹ + ⋯ + k1A + k0I
has the eigenvalues
p(λi) = kmλiᵐ + kₘ−₁λiᵐ−¹ + ⋯ + k1λi+k0
where i = 1,⋯,n, and the same eigenvectors as A. That is, show that p(A)ei = p(λi)ei

Answer 1

The Spectral Mapping Theorem states that a polynomial matrix has eigenvalues and eigenvectors related to the original matrix.

Step-by-step explanation:

The Spectral Mapping Theorem states that if Aeᵢ = λᵢeᵢ for i = 1,⋯,n, then the polynomial matrix p(A) = kₘAᵐ + kₘ−₁Aᵐ−¹ + ⋯ + k₁A + k₀I has the eigenvalues p(λi) = kmλiᵐ + kₘ−₁λiᵐ−¹ + ⋯ + k₁λi+k₀ where i = 1,⋯,n, and the same eigenvectors as A. In other words, p(A)ei = p(λi)ei.