[ODE] Eigenvalues polynomial proof.

[harry901 (蛋笨是的唸來過倒)]

Let L_k(λ) be the polynomial in λ of degree n-1 defined by the eq
          n    λ-λ_j
L_k(λ) = Π (---------)     where λ_1,λ_2,...,λ_n are n distinct scalars.
         j=1  λ_k-λ_j
         j≠k

(a) Prove that
               0  if λ_i≠λ_k
    L_k(λ_i)={1  if λ_i =λ_k

(b) Let y_1,...,y_n be n arbitrary scalars, and let
            n
    p(λ) = Σ y_kL_k(λ)
            k=1
    Prove that p(λ) is the only polynomial of degree ≦ n-1 which satisfies
    the n equations p(λ_k)=y_k    for k=1,2,....,n.
               n
(c) Prove that Σ L_k(λ)=1 for every λ, and deduce that for every square
               k=1   n
    matrix A we have Σ L_k(A)=I, where I is the identity martix and
                     k=1
           n   A - λ_jI
    L_k(A)=Π(-----------)   for k=1,2,....,n.
          j=1 λ_k - λ_j
          j≠k

--
小武的照片  歡迎來我的相本晃晃喔   ^_^
http://tinyurl.com/d289a
拍的不好請多包含   ^^"

--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 218.169.11.178