Section 35.7 Jordan normal form
What.
A block-diagonal matrix
\begin{equation*}
\inv{P}AP
= \begin{bmatrix} J_1 \\ \amp J_2 \\ \amp \amp \ddots \\ \amp \amp \amp J_t \end{bmatrix}\text{,}
\end{equation*}
where each block J_i is a Jordan block. That is, each J_i is in (lower triangular) scalar-triangular form, with an eigenvalue \lambda down the diagonal, such that J_i - \lambda I is in elementary nilpotent form. There can be multiple blocks corresponding to the same eigenvalue, but all blocks for a specific eigenvalue appear consecutively, ordered by size of the block, largest to smallest.When.
The characteristic polynomial of A factors completely as
\begin{equation*}
c_A(\lambda)
= (\lambda - \lambda_1)^{m_1}
(\lambda - \lambda_2)^{m_2}
\dotsm
(\lambda - \lambda_\ell)^{m_\ell}\text{.}
\end{equation*}
How.
First, obtain a matrix M such that U = \inv{M} A M is in triangular block form. (See summary Section 35.4.) Write U_1,U_2,\dotsc,U_\ell for the blocks in U\text{.} The matrix U_1 corresponds to eigenvalue \lambda_1 of A\text{,} and has size m_1 \times m_1\text{,} where m_1 is the algebraic multiplicity of \lambda_1\text{.} This matrix can be decomposed as
\begin{equation*}
U_1 = \lambda_1 I + N_1 \text{,}
\end{equation*}
where N_1 is an m_1 \times m_1\text{,} upper triangular, nilpotent matrix. Next, obtain a matrix Q_1 such that \inv{Q_1} N_1 Q_1 is in triangular-block nilpotent form. (See summary Section 35.6.)
Now repeat for N_2\text{,} where
\begin{equation*}
U_2 = \lambda_2 I + N_2 \text{,}
\end{equation*}
to obtain Q_2 such that \inv{Q_2} N_2 Q_2 is in triangular-block nilpotent form. Then repeat for N_3\text{,} where
\begin{equation*}
U_3 = \lambda_3 I + N_3 \text{,}
\end{equation*}
obtaining matrix Q_3\text{.} And so on.
Finally, take P = M Q\text{,} where
\begin{equation*}
Q
= \begin{bmatrix} Q_1 \\ \amp Q_2 \\ \amp \amp \ddots \\ \amp \amp \amp Q_\ell \end{bmatrix}\text{.}
\end{equation*}
Result.
Similar to the summary of triangular-block nilpotent form (Section 35.6), the number of blocks corresponding to a particular eigenvalue \lambda is equal to the nullity of A - \lambda I (which is the same as the dimension of the eigenspace E_\lambda(A)). Write m for the algebraic multiplicity of \lambda\text{.} The largest (and first) block corresponding to \lambda has dimensions k \times k\text{,} where k is the smallest positive integer such that
\begin{equation*}
\rank (A - \lambda I)^k = n-m \text{.}
\end{equation*}
(In other words, k is the degree of nilpotency of the nilpotent part of the “macro” triangular block corresponding to \lambda\text{.}) The sizes of the remaining Jordan blocks for \lambda can be deduced from the ranks of the powers
\begin{equation*}
(A - \lambda I)^{k-1}, (A - \lambda I)^{k-2}, \dotsc, (A - \lambda I)^1 \text{.}
\end{equation*}