Chapter 26: real symmetric matrix diagonalizable
https://tex.stackexchange.com/questions/30619/what-is-the-best-symbol-for-vector-matrix-transpose
Theorem 26.1
實對稱矩陣的特徵值皆是實數,且對應特徵向量是實向量。
{{A∈Mn×n(R)real matrixA⊺=Asymmetric matrixreal symmetric matrixAx=λx{λ∈Ccomplex eigenvalue0≠x∈Cncomplex eigenvector⇓{λ∈Rreal eigenvalue(1)x∈Rnreal eigenvector(2)
Proof. (1)
Ax=λx¯A¯x=¯Ax=¯λx=¯λ¯x¯x⊺¯A⊺=(¯A¯x)⊺=(¯λ¯x)⊺=¯λ¯x⊺¯x⊺Asymmetric=¯x⊺A⊺real=¯x⊺A=¯λ¯x⊺λ¯x⊺x=¯x⊺(λx)⋅x=Ax=λx¯x⊺Ax=¯λ¯x⊺xλ¯x⊺x=¯λ¯x⊺x(λ−¯λ)¯x⊺x=0∧{¯x⊺x=n∑i=1|xi|2x≠0⇒¯x⊺x≠0λ−¯λ=0λ=¯λ⇔λ∈R
Proof. (1) fast concept
(¯A¯x)⊺x=(¯x⊺¯A⊺)xsymmetric=(¯x⊺¯A)x=¯x⊺(¯Ax)(L)=(¯A¯x)⊺x=¯x⊺(¯Ax)=(R)(L)=(¯A¯x)⊺xAx=λx=(¯λ¯x)⊺x=¯λ¯x⊺x(R)=¯x⊺(¯Ax)real=¯x⊺(Ax)Ax=λx=¯x⊺(λx)=λ¯x⊺x¯λ¯x⊺x=(¯A¯x)⊺x=¯x⊺(¯Ax)=λ¯x⊺x¯λ¯x⊺x=λ¯x⊺x
Proof. (2)
???
推論特徵空間 N(A−λI) (A−λI 的零空間) 為 Rn 的子空間,故 x∈N(A−λI) 是一個非零實向量。
Theorem 26.2
實對稱矩陣對應相異特徵值的特徵向量互為正交。
{{A∈Mn×n(R)real matrixA⊺=Asymmetric matrixreal symmetric matrixAx=λx???{λ∈Rreal eigenvaluex∈Rnreal eigenvector{Ax1=λ1x1(e1)Ax2=λ2x2(e2)λ1≠λ2⇓x⊺1x2=0⇔x1⊥x2
Proof. (1)
Ax2=λ2x2x⊺1Ax2x⊺1⋅=x⊺1λ2x2=λ2x⊺1x2=(1)Ax1=λ1x1x⊺1A⊺=(Ax1)⊺=(λ1x1)⊺=λ1x⊺1x⊺1A⊺=λ1x⊺1x⊺1Ax2symmetric=x⊺1A⊺x2⋅x2=λ1x⊺1x2=(2)λ2x⊺1x2(1)=x⊺1Ax2(2)=λ1x⊺1x2λ2x⊺1x2=λ1x⊺1x2(λ2−λ1)x⊺1x2=0∧λ1≠λ2x⊺1x2=0
Proof. (1) fast concept
(Ax1)⊺x2=(x⊺1A⊺)x2symmetric=(x⊺1A)x2=x⊺1(Ax2)(L)=(Ax1)⊺x2=x⊺1(Ax2)=(R)(L)=(Ax1)⊺x2(e1)=(λ1x1)⊺x2=λ1x⊺1x2(R)=x⊺1(Ax2)(e2)=x⊺1(λ2x2)=λ2x⊺1x2λ1x⊺1x2=(Ax1)⊺x2=x⊺1(Ax2)=λ2x⊺1x2λ1x⊺1x2=λ2x⊺1x2
Theorem 26.3
{{A∈Mn×n(R)real matrixA⊺=Asymmetric matrixreal symmetric matrixAx1=λx1(e)x⊺2x1=0⇔x2⊥x1(o)⇓Ax2⊥x1⇔(Ax2)⊺x1=0
Proof. (Ax2)⊺x1=(x⊺2A⊺)x1symmetric=(x⊺2A)x1=x⊺2(Ax1)(e)=x⊺2(λx1)=λx⊺2x1(o)=λ⋅0=0(Ax2)⊺x1=0⇔Ax2⊥x1