Math 375 - 11/21

Eigenvalues and Eigenvectors • Definition ○ If T:V→V is linear and V is a vector space ○ Then v∈V is an eignevector of T with eigenvalue λ if § v≠0 § Tv=λv • Theorem ○ Linear transformation T:Rn→Rn (or ℂ^n→ℂn) ○ matrix(T)=T=[■8(t_11&⋯&t_1n@⋮&⋱&⋮@t_n1&⋯&t_nn )] ○ Then λ is an eigenvalue of T if ○ det⁡(T−λI)=0 • Characteristic Polynomial ○ det⁡(T−λI) is the called characteristic polynomial of T ○ f(λ)=det⁡(T−λI)=|■8(t_11−λ&t_12&⋯&t_1n@t_21&t_22−λ&…&t_2n@⋮&⋮&⋱&⋮@t_n1&t_n2&⋯&t_nn−λ)| ○ =(−λ)^n+c_1 (−λ)^(n−1)+…+c_(n−1) (−λ)+c_n • How to Find Eigenvalues ○ Solve det⁡(T−λI)=0 ○ Get roots λ_1,…,λ_n (possibly repeated) • How to Find Eigenvectors ○ Solve (T−λI)v=0 ○ For λ=λ_1,λ=λ_2,…,λ=λ_n ○ (T−λI)v=0 is n equations with n unknowns ○ Typically v=0 is the only solution for some λ=λ_i ○ Then det⁡(T−λI)=0, and there is a solution v≠0 • Coefficients of Characteristic Polynomial ○ By definition § f(λ)=(−λ)^n+c_1 (−λ)^(n−1)+…+c_(n−1) (−λ)+c_n ○ By Fundamental Theorem of Algebra § f(λ)=a(λ_1−λ)(λ_2−λ)⋯(λ_n−λ) ○ Comparing the coefficient of (−λ)^n, we get § a=1 ○ Setting λ=0 to both polynomials we get § c_n=det⁡T=λ_1 λ_2…λ_n ○ By Vieta