So i'm trying to get an idea of what an invariant subspace is and so please let me know if my understanding is correct. Given that you have some vector subspace being a collection of a particular number of vectors with the the space denoted as |[tex]\gamma[/tex]>. If you have some other collection of vectors, not necessairly being a subspace in itself ... however we'll say that this collection of vectors is denoted as |[tex]\beta[/tex]>. If |[tex]\gamma[/tex]> is to be |[tex]\beta[/tex]> invariant, does that mean beta is contained in the vector space gamma? Or for gamma to be beta invariant, does that mean the vector collection |[tex]\beta[/tex]> must make up a vector subspace itself spanning the vector subspace |[tex]\gamma[/tex]>? If this is the definition, must beta be a collection of orthogonal vector states? I'm not sure which definition it should be, or if I'm even right with any of the definitions. Thanks for the responses.