Flag (linear algebra)

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration):

If we write the dim V_i = d_i then we have

where n is the dimension of V (assumed to be finite-dimensional). Hence, we must have k ≤ n. A flag is called a complete flag if d_i = i, otherwise it is called a partial flag.

A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.

The signature of the flag is the sequence (d₁, … d_k).

Under certain conditions the resulting sequence resembles a flag with a point connected to a line connected to a surface.

An ordered basis for V is said to be adapted to a flag if the first d_i basis vectors form a basis for V_i for each 0 ≤ i ≤ k. Standard arguments from linear algebra can show that any flag has an adapted basis.

Any ordered basis gives rise to a complete flag by letting the V_i be the span of the first i basis vectors. For example, the standard flag in Rⁿ is induced from the standard basis (e₁, ..., e_n) where e_i denotes the vector with a 1 in the ith slot and 0's elsewhere. Concretely, the standard flag is the subspaces:

An adapted basis is almost never unique (trivial counterexamples); see below.

A complete flag on an inner product space has an essentially unique orthonormal basis: it is unique up to multiplying each vector by a unit (scalar of unit length, like 1, -1, i). This is easiest to prove inductively, by noting that ${\displaystyle v_{i}\in V_{i-1}^{\perp }<V_{i}}$, which defines it uniquely up to unit.

...
Wikipedia