DLA Triangular-block nilpotent form

Section 35.6 Triangular-block nilpotent form

What.

A block-diagonal matrix

\begin{equation*} \inv{P}AP = \begin{bmatrix} N_1 \\ \amp N_2 \\ \amp \amp \ddots \\ \amp \amp \amp N_s \end{bmatrix}\text{,} \end{equation*}

where each block \(N_i\) is in elementary nilpotent form, and the blocks are arranged in order of size, \(N_1\) largest to \(N_s\) smallest.

When.

Matrix \(A\) is nilpotent.

How.

Determine a complete set of independent subspaces of \(\R^n\) (or \(\C^n\text{,}\) as appropriate) that satisfy the following. Each subspace is a cyclic space for \(A\text{,}\) and has an \(A\)-cyclic basis whose last element lies in the null space of \(A\text{.}\) Order these subspaces by dimension, from largest to smallest. Take the first however many columns of \(P\) to be the cyclic basis vectors for the first (largest dimension) cyclic subspace, the next however many columns of \(P\) to be the cyclic basis vectors from the next subspace, and so on.

Result.

The number of blocks, \(s\text{,}\) is equal to the nullity of \(A\text{.}\) The largest block, \(N_1\text{,}\) has dimensions \(k\times k\text{,}\) where \(k\) is the degree of nilpotency of the nilpotent matrix \(A\text{.}\) The sizes of the remaining blocks can be deduced from the ranks of the powers \(A^{k-1}, A^{k-2}, \dotsc, A\text{,}\) as described in Subsection 33.4.2.