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Section 35.6 Triangular-block nilpotent form

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What.

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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*}

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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.

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How.

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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.

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Result.

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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.