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![Question 1 • Question ○ Let θ∈R. ○ Find all eigenvalues and eigenvectors of the following matrix ○ A=[■8(cosθ&−sinθ@sinθ&cosθ )] • Answer ○ |A−λI|=|■8(cosθ−λ&−sinθ@sinθ&cos〖θ−λ〗 )|=\ (cosθ−λ)^2+sin^2θ=0 ○ ⇒λ^2−(2 cosθ )λ+1=0 ○ ⇒λ=cosθ±isinθ ○ When λ_1=cosθ−isinθ § [■8(i sinθ&−sinθ@sinθ&i sinθ )][█(x_1@x_2 )]=0 § {█(i sinθ x_1−sinθ x_2=0@sinθ x_1+i sinθ x_2=0)┤⇒ix_1=x_2 § ⇒v_1=t(1,i), t∈ℂ ○ When λ_2=cosθ+isinθ § [■8(−i sinθ&−sinθ@sinθ&−i sinθ )][█(x_1@x_2 )]=0 § {█(−i sinθ x_1−sinθ x_2=0@sinθ x_1−i sinθ x_2=0)┤⇒−ix_1=x_2 § ⇒v_1=t(1,−i), t∈ℂ Question 2 • Question ○ Let V be a vector space and let T:V→V be a linear map ○ Suppose x∈V is an eigenvector for T with eigenvalue λ. ○ Prove that, for each polynomial, ○ the linear map P(T) has eigenvector x with eigenvalue P(λ) • Answer ○ Let P(λ)=c_n λ^n+c_(n−1) λ^(n−1)+…+c_1 λ+c_0 ○ (P(T))(x) ○ =(c_n T^n+c_(n−1) T^(n−1)+…+c_1 T+c_0 )(x) ○ =c_n T^n (x)+c_(n−1) T^(n−1) (x)+…+c_1 T(x)+c_0 x ○ =c_n λ^n x+c_(n−1) λ^(n−1) x+…+c_1 λx+c_0 x ○ =(c_n λ^n+c_(n−1) λ^(n−1)+…+c_1 λ+c_0 )x ○ =(P(λ))x Question 3 • Given ○ Let V be a vector space and let T:V→V be a linear map ○ Let c be a scalar. ○ Suppose T^2 has an eigenvalue c^2 • Prove ○ T has either c or −c as an eigenvalue • Proof ○ ∃x∈V, ≠0 ○ (T^2−c^2 I)x=0 ○ (T+cI)[(T−cI)x]=0 ○ When (T−cI)x≠0 § (T−cI)x is a eigenvector for T with eigenvalue of −c ○ When (T−cI)x=0 § x is a eigenvector for T with eigenvalue of c Question 4 • Given ○ Let V be a vector space and let T:V→V be a linear map ○ Suppose x,y∈V are eigenvectors of T with eigenvalues λ and μ. • Prove ○ If ax+by (a,b∈R) is an eigenvector of T, ○ then a=0 or b=0 or λ=μ • Proof](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a2b72b31b194.png)
![Theorem • V has a basis v_1,…,v_n, and another basis w_1,…,w_n • Let T be a linear transformation V→V • Define the following matrices ○ A≔matrix(T,v_i ) ○ B≔matrix(T,w_i ) ○ C≔∀i∈{1,…,n}, C(w_i )=v_i • Then B=C^(−1) AC Question • Given ○ T:R3→R3 ○ f(T)=(2−λ)^2 (3−λ) ○ dim(Null(T−2I))=1 • Find T ○ T=[■8(2&1&0@∗&2&0@0&∗&3) ○ For λ=2 ○ Tv=2v ○ ⇒v=k[█(1@0@0)]](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a2b72ef9c5a8.png)
![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→ℂ^n) ○ 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=detT=λ_1 λ_2…λ_n ○ By Vieta](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a2b731f3f050.png)
![Eigenvalues and Eigenvectors • Definition ○ T:V→V linear, for {█(x∈V@λ∈ℂ)┤, (x≠0) ○ We say x is an eigentvector for T with eigenvalue λ if Tx=λx • Note ○ Tx=λx ○ ⇒Tx−λx=0 ○ ⇒(T−λI)x=0 ○ ⇒x∈N(T−λI) Find all eigenvalues and eigenvectors • T=I ○ Tx=1x, ∀x∈V ○ Eigenvalue = 1 with eigenvectors of all elements in V • T=0 ○ Tx=0x, ∀x∈V ○ Eigenvalue = 0 with eigenvectors of all elements in V • T=[■(c_1&&@&⋱&@&&c_n )], (c_i≠c_j for i=j) ○ [■(c_1&&@&⋱&@&&c_n )] e_i=c_i e_i ○ Eigenvalue = c_i with eigenvector of te_i, (t∈R, t≠0) • T=[■8(1&2@2&1) ○ det(T−λI)=0 § |■8(1−λ&2@2&1−λ)|=0 § (λ−3)(λ+1)=0 ○ λ=3 § [■8(1−3&2@2&1−3)][█(x@y)]=[█(0@0)]⇒x=y § Eigenvector: [█(t@t)](t∈R, t≠0) ○ λ=−1 § [■8(2&2@2&2)][█(x@y)]=[█(0@0)]⇒x+y=0 § Eigenvector: t[█(1@−1)](t∈R, t≠0) • T=[■8(0&−1@1&0) ○ det(T−λI)=0 § |■8(−λ&−1@1&−λ)|=0 § λ^2+1=0 ○ λ=i § [■8(−i&−1@1&−i)][█(x@y)]=[█(0@0)]⇒y=−ix § Eigenvector: t[█(1@−i)](t∈ℂ, t≠0) ○ λ=−i § [■8(i&−1@1&i)][█(x@y)]=[█(0@0)]⇒y=ix § Eigenvector: t[█(1@i)](t∈ℂ, t≠0) Multiplicity of Eigenvalues • T=[■(3&&@&3&@&&4)] ○ Eigenvalues: λ=3 or λ=4 ○ dim〖N(T−λI)={■8(2&λ=3@1&λ=4@0&otherwise)┤〗](https://shawnzhong.com/wp-content/uploads/2017/12/img_5a2b7365d72ee.png)
