设n×nn\times nn×n矩阵有nnn个线性无关的特征向量x1,...,xnx_1,...,x_nx1​,...,xn​，令S=(x1,...,xn)S =(x_1,...,x_n)S=(x1​,...,xn​)，则：

AS=A(x1,...,xn)=(λ1x1,...,λnxn)=(x1,...,xn)(λ1...λn)AS = A(x_1,...,x_n) = (\lambda _1 x_1,...,\lambda_n x_n ) = (x_1,...,x_n)\begin{pmatrix}\lambda_1 & & \\\\ & ... & \\\\ & & \lambda_n\end{pmatrix}AS=A(x1​,...,xn​)=(λ1​x1​,...,λn​xn​)=(x1​,...,xn​)⎝⎜⎜⎜⎜⎛​λ1​​...​λn​​⎠⎟⎟⎟⎟⎞​

AS=SΛ⇒S−1AS=ΛAS = S\Lambda \Rightarrow S^{-1}AS = \LambdaAS=SΛ⇒S−1AS=Λ A