PlanetMath: flag

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flag

(Definition)

Let be a finite-dimensional vector space. A filtration of subspaces

$\displaystyle V_1\subset V_2\subset\cdots \subset V_n= V$

is called a flag in

. We speak of a complete flag when

$\displaystyle \dim V_i = i$

for each $i=1,\ldots,n$ .

Next, putting

$\displaystyle d_k = \dim V_k,\quad k=1,\ldots n,$

we say that a list of vectors $(u_1,\ldots,u_{d_n})$ 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 $\mathbb{R}^n$ . For each $k=1,\ldots,n$ let

be the span of $e_1,\ldots,e_k$ , where

denotes the $j^{\text{th}}$ basic vector, i.e. the column vector with

in the $j^{\text{th}}$ position and zeros everywhere else. The

give a complete flag in $\mathbb{R}^n$ . The list $(e_1,e_2,\ldots, e_n)$ is an adapted basis relative to this flag, but the list $(e_2,e_1,\ldots,e_n)$ is not.

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.

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"flag" is owned by rmilson. [ full author list (3) ]

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Also defines:

adapted basis, complete flag

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Cross-references: finite dimensional, partially ordered set, chain, column vector, span, characterization, basis, vectors, subspaces, filtration, vector space, finite-dimensional
There are 8 references to this entry.

This is version 6 of flag, born on 2002-06-01, modified 2006-09-23.
Object id is 2994, canonical name is Flag.
Accessed 5005 times total.

Classification:

AMS MSC:	15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
	06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)

Pending Errata and Addenda

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