[行列解析3.1]ジョルダン標準形の定理

J_1(\lambda) = [\lambda], \quad
J_2(\lambda) =
\begin{bmatrix}
\lambda & 1 \\
0 & \lambda
\end{bmatrix}

J = J_{n_1}(\lambda_1) \oplus J_{n_2}(\lambda_2) \oplus \cdots \oplus J_{n_q}(\lambda_q),\\
\quad n_1 + n_2 + \cdots + n_q = n

A = S \left( J_{n_1}(0) \oplus J_{n_2}(0) \oplus \cdots \oplus J_{n_m}(0) \right) S^{-1}

S_1^{-1} A_1 S_1 = 
\begin{bmatrix}
J_{k_1} & 0 \\
0 & J
\end{bmatrix}

\begin{bmatrix}
1 & 0 \\
0 & S_1^{-1}
\end{bmatrix}
A
\begin{bmatrix}
1 & 0 \\
0 & S_1
\end{bmatrix}
=
\begin{bmatrix}
0 & a^T S_1 \\
0 & S_1^{-1} A_1 S_1
\end{bmatrix}

\begin{bmatrix}
0 & (a_1^T (I - J_{k_1}^T J_{k_1})) & a_2^T \\
0 & J_{k_1} & 0 \\
0 & 0 & J
\end{bmatrix}

\begin{bmatrix}
0 & (a_1^T e_1)e_1^T & a_2^T \\
0 & J_{k_1} & 0 \\
0 & 0 & J
\end{bmatrix}

\begin{bmatrix}
J_{k_1} & 0 & 0 \\
0 & 0 & a_2^T \\
0 & 0 & J
\end{bmatrix} 

A = S
\begin{bmatrix}
J_{n_1}(\lambda_1) & & \\
 & \ddots & \\
 & & J_{n_q}(\lambda_q)
\end{bmatrix}
S^{-1}

\begin{aligned}
& \operatorname{rank}(A - \lambda I)^k \\
& = \operatorname{rank}(J - \lambda I)^k \\
& = \operatorname{rank} J_{m_1}(0)^k +  \\
& \quad \quad \quad \cdots + \operatorname{rank} J_{m_p}(0)^k + \operatorname{rank}(\hat{J} - \lambda I)^k \\
&= \operatorname{rank} J_{m_1}(0)^k + \cdots + \operatorname{rank} J_{m_p}(0)^k + m \\
& \quad \text{for} \quad k=1,2,\cdots
\end{aligned}

\operatorname{rank} J_\ell(0)^{k-1} - \operatorname{rank} J_\ell(0)^k \\
 =
\begin{cases}
1 & \text{if } k \leq \ell \\
0 & \text{if } k \gt \ell
\end{cases}, \\
\quad k = 1,2,\ldots

\begin{align}
& r_k(A,\lambda) = \operatorname{rank}(A - \lambda I)^k, \notag \quad \\
&  r_0(A,\lambda) := n \notag
\end{align}

\begin{align}
& w_k(A,\lambda) = r_{k-1}(A,\lambda) - r_k(A,\lambda), \notag \\
& w_1(A,\lambda) := n - r_1(A,\lambda) \notag
\end{align}

\begin{align}
J = 
& J_3(0) \oplus J_3(0) \notag \\
& \quad \oplus J_2(0) \oplus J_2(0) \oplus J_2(0) \notag \\
& \quad \quad \oplus J_1(0) \notag
\end{align}

\begin{align}
&w_k(A,\lambda) \notag \\
& =
\bigl(\operatorname{rank} J_{m_1}(0)^{k-1} - \operatorname{rank} J_{m_1}(0)^k \bigr) + \notag \\
& \quad \quad \cdots +
 \bigl(\operatorname{rank} J_{m_p}(0)^{k-1} - \operatorname{rank} J_{m_p}(0)^k \bigr) \notag \\
& = (1 \text{ if } m_1 \geq k) + \cdots + (1 \text{ if } m_p \geq k) \notag \\
& = \left( \begin{aligned} &\text{固有値} \lambda \text{を持つサイズ } k \text{以上} \\ &\text{のブロックの個数}\end{aligned} \right) \notag 
\end{align}