Math Solutions

(Solution Library) If A ∈ M_n, S A S^-1=\Lambda=diag;(λ_1, ..., λ_n), and p(t) is a polynomial, show that p(A)=S^-1 p(\Lambda) S and that p(\Lambda)=diag;(p(λ_1),

Question: If \(A \in M_{n}, S A S^{-1}=\Lambda=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right)\), and \(p(t)\) is a polynomial, show that

\(p(A)=S^{-1} p(\Lambda) S\) and that \(p(\Lambda)=\operatorname{diag}\left(p\left(\lambda_{1}\right), \ldots, p\left(\lambda_{n}\right)\right) .\) This provides a simple way

to evaluate \(p(A)\) if one can diagonalize $A .$

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Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


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