3 ms·

What's a "standard flag"? Couldn't find anything about it... I would think "upper triangular" would be a weird/bad notion to talk about in a basis-independent

• xanderlewis 2y ago
  A ‘flag’ is just a funny name for an increasing (and therefore increasing in dimension) sequence of subspaces. The ‘standard flag’ is the sequence spanned by the standard basis vectors: start with {0}, then things of the form (x,0,…), then (x,y,0,…), and so on. * What I’m saying is that an upper triangular matrix preserves each of these subspaces because it sends each basis vector e_i into the span of the e_0, …, e_{i-1}. You’re absolutely right that upper triangularity is basis-dependent and so is somewhat ‘weird’/‘evil’ (in fact, not even well-defined) as a purported property of maps rather than matrices. What I meant to say was ‘triangularisable’ by analogy with ‘diagonalisable’ — matrices which represent such a map in some basis. Given a linear operator, its matrix representation is triangular when expressed in a basis B if and only if it preserves the standard flag in basis B. * https://en.wikipedia.org/wiki/Flag_(linear_algebra) https://en.wikipedia.org/wiki/Flag_(linear_algebra)