已知A为n阶方阵且A^2=A,求A的全部特征值.已知矩阵A=-110-2204X1能对角化,求X并计算A^n(n>=1)

Question

已知A为n阶方阵且A^2=A,求A的全部特征值.
已知矩阵A=-1 1 0
-2 2 0
4 X 1
能对角化,求X并计算A^n(n>=1)

Answer 1

1.设 a为矩阵A的特征值,X为对应的非零特征向量.则有 AX = aX.aX = AX = A^2X = A(AX) = A(aX) = aAX = a(aX) = a^2X,(a^2 - a)X = 0,因X为非零向量,所以.0 = a^2 - a = a(a-1),a = 0或1.2.|A-λE|=|-1-λ 1 0 || -2 ...