Section 35.4 Triangular-block form
What.
A block-diagonal form matrix
\begin{equation*}
\inv{P}AP
= \begin{bmatrix} U_1 \\ \amp U_2 \\ \amp \amp \ddots \\ \amp \amp \amp U_\ell \end{bmatrix}\text{,}
\end{equation*}
where each block \(U\) corresponds to a specific eigenvalue \(\lambda\) of \(A\text{,}\) has size equal to the algebraic multiplicity of \(\lambda\text{,}\) and is in scalar-triangular form with \(\lambda\) down the diagonal.
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.
Follow the scalar-triangular form procedure (Procedure 29.4.1) with \(\lambda = \lambda_1\text{,}\) but stop when you have \(m_1\) linearly independent vectors. Use these vectors as the first columns of \(P\text{.}\) Repeat with \(\lambda = \lambda_2\) to get the next \(m_2\) columns of \(P\text{.}\) And so on.
Result.
Each block \(U_j\) will be as in the described result for Scalar-triangular form.