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。

{{AMn×n(R)real matrixA=Asymmetric matrixreal symmetric matrixAx=λx{λCcomplex eigenvalue0xCncomplex eigenvector{λRreal eigenvalue(1)xRnreal eigenvector(2)

Proof. (1)

Ax=λx¯A¯x=¯Ax=¯λx=¯λ¯x¯x¯A=(¯A¯x)=(¯λ¯x)=¯λ¯x¯xAsymmetric=¯xAreal=¯xA=¯λ¯xλ¯xx=¯x(λx)x=Ax=λx¯xAx=¯λ¯xxλ¯xx=¯λ¯xx(λ¯λ)¯xx=0{¯xx=ni=1|xi|2x0¯xx0λ¯λ=0λ=¯λλR

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¯xx(R)=¯x(¯Ax)real=¯x(Ax)Ax=λx=¯x(λx)=λ¯xx¯λ¯xx=(¯A¯x)x=¯x(¯Ax)=λ¯xx¯λ¯xx=λ¯xx

Proof. (2)

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推論特徵空間 N(AλI) (AλI 的零空間) 為 Rn 的子空間,故 xN(AλI) 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。

{{AMn×n(R)real matrixA=Asymmetric matrixreal symmetric matrixAx=λx???{λRreal eigenvaluexRnreal eigenvector{Ax1=λ1x1(e1)Ax2=λ2x2(e2)λ1λ2x1x2=0x1x2

Proof. (1)

Ax2=λ2x2x1Ax2x1=x1λ2x2=λ2x1x2=(1)Ax1=λ1x1x1A=(Ax1)=(λ1x1)=λ1x1x1A=λ1x1x1Ax2symmetric=x1Ax2x2=λ1x1x2=(2)λ2x1x2(1)=x1Ax2(2)=λ1x1x2λ2x1x2=λ1x1x2(λ2λ1)x1x2=0λ1λ2x1x2=0

Proof. (1) fast concept

(Ax1)x2=(x1A)x2symmetric=(x1A)x2=x1(Ax2)(L)=(Ax1)x2=x1(Ax2)=(R)(L)=(Ax1)x2(e1)=(λ1x1)x2=λ1x1x2(R)=x1(Ax2)(e2)=x1(λ2x2)=λ2x1x2λ1x1x2=(Ax1)x2=x1(Ax2)=λ2x1x2λ1x1x2=λ2x1x2

Theorem 26.3

{{AMn×n(R)real matrixA=Asymmetric matrixreal symmetric matrixAx1=λx1(e)x2x1=0x2x1(o)Ax2x1(Ax2)x1=0

Proof. (Ax2)x1=(x2A)x1symmetric=(x2A)x1=x2(Ax1)(e)=x2(λx1)=λx2x1(o)=λ0=0(Ax2)x1=0Ax2x1