DLA Jordan normal form

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*}