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Section 33.2 Terminology and notation

triangular-block nilpotent form

a matrix in block diagonal form where each block is in elementary nilpotent form

Remark 33.2.1.

When determining a transition matrix to take a matrix to block diagonal form, the order of the blocks depends on the order in which the bases for the associated invariant subspaces are placed as the columns in the transition matrix. This implies that two block diagonal matrices that have the same blocks but in different orders are similar. Since elementary nilpotent form is so simple, the only thing that distinguishes one block from another in triangular-block nilpotent form is the size of the blocks.

In order to help settle on a definitive universal form to answer Question 28.1.1 that can be used to compare whether matrices are similar or not (without having to attempt to construct a transition matrix), we arbitrarily adopt the convention that a matrix in triangular-block nilpotent form is assumed to have blocks that appear in order of descending size.