質問: 実対称行列Aを対角化するような直行行列を一つ求めよ
回答: A= 1,4,-2 4,1,2 -2,2,4 |A-λI|=0 |1-λ、4、-2|=0 |4,1-λ、2| |-2,2,4-λ| λ=-4,5,5 λ=-4 (5,4,-2)~(10-2) (4,5,2)...(012) (-2,2,8)...(000) x3=1 x1-2=0 x1=2 x2+2=0 x2=-2 V1= 2 -2 1 λ=5 (-4,4,-2)~(2,-2,1) (4,-4,2)...(000) (-2,2,-1)...(000) x3=2 2x1-2x2+2=0 x2=2 2x1-4+2=0 x1=1 V2= 1 2 2 λ=5 (-4,4,-2)~(2,-2,1) (4,-4,2)...(000) (-2,2,-1)...(000) x3=2 2x1-2x2+2=0 x2=-1 x1=-2 V3= -2 -1 2 P= 2/3,1/3,-2/3 -2/3,2/3,-1/3 1/3,2/3,2/3 P-1=P’= 2/3,-2/3,1/3 1/3,2/3,2/3 -2/3,-1/3,2/3 D=P’AP= -4,0,0 0,5,0 0,0,5