固有値

\[ \begin{aligned} &i, j \in \{ 1, \cdots , n \}, \quad a_{ij} \in \mathbb{R} \\ \\ &A = \left( \begin{array}{ccccc} a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & & \vdots & & \vdots \\ a_{n1} & \cdots & a_{nj} & \cdots & a_{nn} \end{array} \right) = \left( a_{ij} \right) \in \mathbb{R}^{n \times n} \\ &: n \text{-dim matrix} \end{aligned} \]

\[ \begin{aligned} &I = \left( \begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & & \ddots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{array} \right) \in \mathbb{R}^{n \times n} , \quad &\boldsymbol{0} = \left( \begin{array}{c} 0 \\ \vdots \\ 0 \end{array} \right) \in \mathbb{R} ^ {n} \end{aligned} \]

\[ \begin{aligned} &\lambda \in \mathbb{C}, \quad &\boldsymbol{v} = \left( \begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array} \right) \in \mathbb{R} ^ {n} , \quad \boldsymbol{v} \neq \boldsymbol{0} \\ \\ &A \boldsymbol{v} = \lambda \boldsymbol{v} \end{aligned} \]

\[ \begin{aligned} (A - \lambda I) \boldsymbol{v} = \boldsymbol{0} \end{aligned} \]

\[ \begin{aligned} &K : \text{field} \\ &V : \text{vector space over } K \\ &T : V \rightarrow V : \text{linear transformation} \end{aligned} \]

\[ \begin{aligned} &\alpha \in K , \quad \boldsymbol{v} \in V \\ &T \boldsymbol{v} = \alpha \boldsymbol{v} \end{aligned} \]

\[ \begin{aligned} &A, B \in \mathbb{R}^{n \times n} : n \text{-dim matrix} \\ &\boldsymbol{v} \in \mathbb{R}^{n} : n \text{-dim vector} \\ &\lambda \in \mathbb{R} : \text{number} \\ \end{aligned} \]

\[ \begin{aligned} &A \boldsymbol{v} = \lambda \boldsymbol{v} \end{aligned} \]

\[ \begin{aligned} &A \boldsymbol{v} = \lambda B \boldsymbol{v} \end{aligned} \]

\[ \begin{aligned} &{}^{\exists} B ^ {-1} \in \mathbb{R}^{n \times n} : n \text{-dim matrix} \\ \Longrightarrow & B ^ {-1} A \boldsymbol{v} = \lambda B ^ {-1} B \boldsymbol{v} \\ \Longleftrightarrow & B ^ {-1} A \boldsymbol{v} = \lambda \boldsymbol{v} \end{aligned} \]

\[ \begin{aligned} &A \in M_{n, n}(\mathbb{R}) \end{aligned} \]

\[ \begin{aligned} &^\forall j = 1, \ldots , n \\ &\lambda_{j} \in \mathbb{C} , \quad \boldsymbol{v}_{j} \in \mathbb{R} ^n , \quad A \boldsymbol{v}_{j} = \lambda_{j} \boldsymbol{v}_{j} \end{aligned} \]

\[ \begin{aligned} A \left( \begin{array}{c} \boldsymbol{v}_{1} \\ \vdots \\ \boldsymbol{v}_{j} \\ \vdots \\ \boldsymbol{v}_{n} \end{array} \right) ^\mathrm{T} &= \left( \begin{array}{c} A \boldsymbol{v}_{1} \\ \vdots \\ A \boldsymbol{v}_{j} \\ \vdots \\ A \boldsymbol{v}_{n} \end{array} \right) ^\mathrm{T} = \left( \begin{array}{c} \lambda_{1} \boldsymbol{v}_{1} \\ \vdots \\ \lambda_{j} \boldsymbol{v}_{j} \\ \vdots \\ \lambda_{n} \boldsymbol{v}_{n} \end{array} \right) ^\mathrm{T} \\ &= \left( \begin{array}{c} \boldsymbol{v}_{1} \\ \vdots \\ \boldsymbol{v}_{j} \\ \vdots \\ \boldsymbol{v}_{n} \end{array} \right) ^\mathrm{T} \left( \begin{array}{ccccc} \lambda_{1} & 0 & \cdots & 0 & 0 \\ 0 & \lambda_{2} & \ddots & & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & & \ddots & \lambda_{n-1} & 0 \\ 0 & 0 & \cdots & 0 & \lambda_{n} \end{array} \right) \end{aligned} \]

\[ \begin{aligned} &V = \left( \begin{array}{c} \boldsymbol{v}_{1} \\ \vdots \\ \boldsymbol{v}_{j} \\ \vdots \\ \boldsymbol{v}_{n} \end{array} \right) ^\mathrm{T} \\ \\ &\Lambda = \left( \begin{array}{ccccc} \lambda_{1} & 0 & \cdots & 0 & 0 \\ 0 & \lambda_{2} & \ddots & & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & & \ddots & \lambda_{n-1} & 0 \\ 0 & 0 & \cdots & 0 & \lambda_{n} \end{array} \right) \end{aligned} \]

\[ \begin{aligned} A V = V \Lambda \end{aligned} \]

\[ \begin{aligned} &A V = V \Lambda \\ \Longleftrightarrow &A = V \Lambda V ^{-1} \\ \Longleftrightarrow &V ^{-1} A V = \Lambda \end{aligned} \]

\[ \begin{aligned} &A V = V \Lambda \\ \Longleftrightarrow &A = V \Lambda V ^{-1} \end{aligned} \]

\[ \begin{aligned} V ^{-1} A V = \Lambda \end{aligned} \]