Flag (linear algebra)

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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):

${\displaystyle \{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.}$

The term flag is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.^[1]

If we write that dimV_i = d_i then we have

${\displaystyle 0=d_{0}<d_{1}<d_{2}<\cdots <d_{k}=n,}$

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