原文恕刪~
Q1:prove that the eigenvalue of a real symmetric matrix must be real
Pf:
假設入為A的特徵根，則存在x不等於0使得Ax=入x
T T T 2
=> x A x = x 入 x = 入 x x = 入 || x ||
另一方面
T T T T T T 2
x A x = x A x = (Ax) x = (入x) = 入 x x = 入 || x ||
2 2 2
入 || x || = 入 || x || 且 || x || 不等於0
所以入屬於實數 !
Q2:Let A be a real symmetric matric. Then eigenvector associated with distinct
eigenvalues are orthogonal.
Pf:
假設x_1、x_2分別為A相對於入1、入2的特徵向量，且入1不等於入2
=> Ax_1=入x_1 ，Ax_2=入x_2
題目所示之Distinct eigenvalues are orthogonal 即証 < x_1 , x_2 > = 0
入_1 < x_1 , x_2 > = < 入_1 x_1 , x_2 > = < Ax_1 , x_2 >
T
= < x_1 , A x_2 >
= < x_1 , A x_2 >
= < x_1 , 入_2 x_2 >
= 入_2 < x_1 , x_2 >
=> (入_1 - 入_2 ) < x_1 , x_2 > = 0
因為入1不等於入2，所以 (入_1 - 入_2 )不等於0
故得< x_1 , x_2 > = 0
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