flag
Let be a finite-dimensional vector space. A filtration
of
subspaces
is called a flag in . We speak of a complete flag when
for each .
Next, putting
we say that a list of vectors is an adapted basis relative to the flag, if the first vectors give a basis of , the first vectors give a basis of , etc. Thus, an alternate characterization of a complete flag, is that the first elements of an adapted basis are a basis of .
Example
Let us consider . For each let be the
span of , where denotes the basic
vector, i.e. the column vector with in the position and
zeros everywhere else. The give a complete flag in .
The list is an adapted basis relative to this
flag, but the list is not.
Generalizations.
More generally, a flag can be defined as a maximal chain in a partially ordered set. If one considers the poset consisting of subspaces of a (finite dimensional) vector space, one recovers the definition given above.
Title | flag |
---|---|
Canonical name | Flag |
Date of creation | 2013-03-22 12:42:35 |
Last modified on | 2013-03-22 12:42:35 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 9 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 06A06 |
Classification | msc 15A03 |
Defines | adapted basis |
Defines | complete flag |