"Stable" Subspace

J.F.N

For a vector space V, what is a "stable" subspace? Also, if T is a linear operator on V, what does it mean for a subspace W of V to be "T-stable"?

No definitions on the web, unfortunately.

Reply 1

evariste_3086

1

J.F.N

For a vector space V, what is a "stable" subspace? Also, if T is a linear operator on V, what does it mean for a subspace W of V to be "T-stable"?

No definitions on the web, unfortunately.

im taking a stab here so if not what you have in mind i apologise. Long time since done all this stuff.

Let V be a vector space.
let W be a sub-space of V
let T be in End(V)
W is stable if T(W)CW for all T in end(V)
W is T-stable if T(W)CW

Reply 2

RichE

15

evariste

im taking a stab here so if not what you have in mind i apologise. Long time since done all this stuff.

Let V be a vector space.
let W be a sub-space of V
let T be in End(V)
W is stable if T(W)CW for all T in end(V)
W is T-stable if T(W)CW

The last of those is usually refered to as T-invariant.

The next-to-last one, that of "stable", would result in only 0 and V being "stable" subspaces.

I must admit I've not come across the term in this fashion.

EDIT: having looked around on the internet it seems that Evariste is right and that "T-stable" is another term for what I was calling "T-invariant" above.

Reply 3

evariste_3086

1

RichE

The last of those is usually refered to as T-invariant.

The next-to-last one, that of "stable", would result in only 0 and V being "stable" subspaces.

well said it been long time :smile:

Found this after search. It seems i need a subset of End(V) and not all of it.

http://perso.wanadoo.fr/eric.chopin/epreuve_ENS_LYON_90_en.htm