Math 375 - 11/27

Question 1 • Question ○ Let θ∈R. ○ Find all eigenvalues and eigenvectors of the following matrix ○ A=[■8(cos⁡θ&−sin⁡θ@sin⁡θ&cos⁡θ )] • Answer ○ |A−λI|=|■8(cos⁡θ−λ&−sin⁡θ@sin⁡θ&cos⁡〖θ−λ〗 )|=\ (cos⁡θ−λ)^2+sin2⁡θ=0 ○ ⇒λ^2−(2 cos⁡θ )λ+1=0 ○ ⇒λ=cos⁡θ±isin⁡θ ○ When λ_1=cos⁡θ−isin⁡θ § [■8(i sin⁡θ&−sin⁡θ@sin⁡θ&i sin⁡θ )][█(x_1@x_2 )]=0 § {█(i sin⁡θ x_1−sin⁡θ x_2=0@sin⁡θ x_1+i sin⁡θ x_2=0)┤⇒ix_1=x_2 § ⇒v_1=t(1,i), t∈ℂ ○ When λ_2=cos⁡θ+isin⁡θ § [■8(−i sin⁡θ&−sin⁡θ@sin⁡θ&−i sin⁡θ )][█(x_1@x_2 )]=0 § {█(−i sin⁡θ x_1−sin⁡θ x_2=0@sin⁡θ x_1−i sin⁡θ 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−c2 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