Eigenvalues and eigenvectors
Three-dimensional matrix example with complex eigenvalues

Consider the cyclic permutation matrix A =. {\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}

This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − λ, whose roots are λ 1 = 1 λ 2 = − 1 2 + i 3 2 λ 3 = λ 2 ∗ = − 1 2 − i 3 2 {\displaystyle {\begin{aligned}\lambda _{1}&=1\\\lambda _{2}&=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}\\\lambda _{3}&=\lambda _{2}^{*}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}\end{aligned}}} where i {\displaystyle i} is an imaginary unit with i 2 = − 1 {\displaystyle i^{2}=-1}.

For the real eigenvalue λ = 1, any vector with three equal nonzero entries is an eigenvector. For example, A = = 1 ⋅. {\displaystyle A{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}5\\5\\5\end{bmatrix}}=1\cdot {\begin{bmatrix}5\\5\\5\end{bmatrix}}.}

For the complex conjugate pair of imaginary eigenvalues, λ 2 λ 3 = 1, λ 2 2 = λ 3, λ 3 2 = λ 2. {\displaystyle \lambda _{2}\lambda _{3}=1,\quad \lambda _{2}^{2}=\lambda _{3},\quad \lambda _{3}^{2}=\lambda _{2}.}

Then A = = λ 2 ⋅, {\displaystyle A{\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}}={\begin{bmatrix}\lambda _{2}\\\lambda _{3}\\1\end{bmatrix}}=\lambda _{2}\cdot {\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}},} and A = = λ 3 ⋅. {\displaystyle A{\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}={\begin{bmatrix}\lambda _{3}\\\lambda _{2}\\1\end{bmatrix}}=\lambda _{3}\cdot {\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}.}

Therefore, the other two eigenvectors of A are complex and are v λ 2 = T {\displaystyle \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}1&\lambda _{2}&\lambda _{3}\end{bmatrix}}^{\textsf {T}}} and v λ 3 = T {\displaystyle \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}} with eigenvalues λ and λ, respectively. The two complex eigenvectors also appear in a complex conjugate pair, v λ 2 = v λ 3 ∗. {\displaystyle \mathbf {v} _{\lambda _{2}}=\mathbf {v} _{\lambda _{3}}^{*}.}
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