Math 375 – 11/21 Nov 21, 2017 Shawn Math 375 No comments yet Math 375 – 11/20 Nov 21, 2017 Shawn Math 375 No comments yet Math 431 – 9/11 Nov 21, 2017 Shawn Math 375 No comments yet
![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)
![Probability Space • Ω: sample space (list of all outcomes) • F: collection of events (subsets of Ω) • P: probability measure ○ P(A)∈[0,1] ○ P(∅)=0 ○ P(Ω)=1 ○ For disjoint A_1,A_2,…:P(⋃24_(i=1)^∞▒A_i )=∑_(i=1)^∞▒PA_i ) Equally Likely Outcome • P(ω)=1/(#Ω),∀ω∈Ω • P(A)=(#A)/(#Ω) • Example: 431 game with full deck ○ Ω={(c_1,c_2,c_3 )│■8(c_1 is my card@c_2 is your first card@c_3 is your second card@and they are all distinct)} ○ P(A)=(#A)/(#Ω) ○ W_7={(c_1,c_2,c_3 )∈Ω| (c_2≥7 and c_2c_1 ) or (c_27 and c_3c_1 )} ○ #Ω=52×51×50=(52)_3 ○ Note: (n)_k=n!/(n−k)! • Example: 431 game with replacement ○ Ω={(c_1,c_2,c_3 )│■8(c_1 is my card@c_2 is your first card@c_3 is your second card)} ○ W_7={(c_1,c_2,c_3 )∈Ω| (c_2≥7 and c_2c_1 ) or (c_27 and c_3c_1 )} ○ #Ω=〖52〗^3 Different Types of Random Experiments • S={1,…,n} • Sampling with replacement where order matters ○ Ω=S^k={(s_1,…,s_k )|s_i∈S} ○ #Ω=n^k • Sampling without replacement where order matters ○ Ω={(s_1,…,s_k )|s_i∈S and ∀i≠j:s_i≠s_j } ○ #Ω=n(n−1)⋯(n−k+1)=n!/(n−k)!=(n)_k • Sampling without replacement where order is irrelevant ○ Ω={A⊆S|#A=k}](https://shawnzhong.com/wp-content/uploads/2017/11/img_5a13a28c17868.png)
