Skip to main content

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.