Section 32.3 Terminology and notation
- elementary nilpotent form
a lower triangular matrix that is all zeros except for a line of ones on the first sub-diagonal
Remark 32.3.1.
An upper triangular matrix that is all zeros except for a line of ones on the first super-diagonal could also be considered to be in elementary nilpotent form, but we will focus on the lower triangular version because it makes the similarity pattern more obvious.
The next two definitions are relative to a specific \(n \times n\) matrix \(A\text{,}\) and can be applied to vectors and subspaces in \(\R^n\) or in \(\C^n\text{,}\) as appropriate.
- \(A\)-cyclic subspace
a subspace of the form \(\Span \{ \uvec{w}, A \uvec{w}, A^2 \uvec{w}, \dotsc \}\) for some vector \(\uvec{w}\)
If we begin with the matrix \(A\) and a single vector \(\uvec{w}\text{,}\) then we say that
is the \(A\)-cyclic subspace generated by \(\uvec{w}\). On the other hand, if we begin with the matrix \(A\) and a subspace \(W\text{,}\) and are able to identify a vector \(\uvec{w}\) in \(W\) which generates \(W\) as an \(A\)-cyclic subspace, then we say that \(\uvec{w}\) is an \(A\)-cyclic vector for \(W\).