特征值与特征向量 f(λ)=akλk+⋯+a1λ+a0=0f(\lambda)=a_k\lambda^k + \cdots + a_1\lambda + a_0 = 0f(λ)=akλk+⋯+a1λ+a0=0 若a0=0a_0=0a0=0则 0 是f(λ)f(\lambda)f(λ)的根 若ak+ak−1+⋯+a1+a0=0a_k+a_{k-1}+\cdots+a_1+a_0=0ak+ak−1+⋯+a1+a0=0则 1 是f(λ)f(\lambda)f(λ)的根 若偶次项系数和等于奇次项系数之和则-1 是f(λ)f(\lambda)f(λ)的根 aia_iai都是整数，则f(λ)f(\lambda)f(λ)的有理根都是整数且均是a0a_0a0的因子 (λE−A)ξ=0(\lambda E- A)\xi = 0(λE−A)ξ=0 (kλE−kA)ξ=0(k\lambda E - kA)\xi = 0(kλE−kA)ξ=0 (λkE−Ak)ξ=0(\lambda^k E - A^k)\xi = 0(λkE−Ak)ξ=0 (f(λ)E−f(A))ξ=0(f(\lambda) E - f(A))\xi = 0(f(λ)E−f(A))ξ=0 (Eλ−A−1)ξ=0(\frac{E}{\lambda} - A^{-1})\xi = 0(λE−A−1)ξ=0 (∣A∣λE−A∗)ξ=0(\frac{|A|}{\lambda}E - A^*)\xi = 0(λ∣A∣E−A∗)ξ=0 (λE−P−1AP)P−1ξ=0(\lambda E - P^{-1}AP)P^{-1}\xi=0(λE−P−1AP)P−1ξ=0