📂Linear Algebra

What is a Flag in Linear Algebra?

Definition^{1 2}

$n$Dimension Vector space $V$Subspaces sequences $\left\{ W_{i} \right\}$ satisfying the following equations are termed flags.

$$\left\{ \mathbf{0} \right\} = W_{0} \lneq W_{1} \lneq W_{2} \lneq \cdots \lneq W_{k-1} \lneq W_{k} = V$$

By definition, the following holds.

$$0 = \dim V_{0} \lt \dim V_{1} \lt \dim V_{2} \lt \cdots \lt \dim V_{k-1} \lt \dim V_{k} = n$$

Explanation

The term flag is used because, at first glance, the equations resemble flags being hoisted. ³

By definition, it is obvious that $k \le n$, and if $\dim V_{i} = i$ (i.e., $k=n$), it is called a complete flag, otherwise a partial flag.

When $d_{i} = \dim V_{i}$, the sequence $\left\{ d_{i} \right\}$ is known as the flag’s signature.

See Also

Filtration

$$A_{1} \subset A_{2} \subset \cdots \subset A_{n} \subset \cdots$$ In mathematics in general, structures that form a Nested Sequence as above are referred to as Filtration.

• Filtration of a Stochastic Process
• Filtration of Complexes