Finding invariant subspaces

1. Feb 23, 2009

samkolb

1. The problem statement, all variables and given/known data

Let V be a finite dimensional, nonzero complex vector space. Let T be be a linear map on V. Show that V contains invariant subspaces of dimension j for j=1, ..., dim V.

2. Relevant equations
Since V is complex, V contains an invariant subspace of dimension 1.

3. The attempt at a solution
I started with dim V=3. Then V contains an invariant subspace of dimension 1.
Let U1=span{u1} denote this space, and extend this to a basis for V: V=span{u1,v2,v3}.

What I would like to do is show that span{v2,v3} contains an invariant subspace of dimension 1, span{u2}. Then form the invariant subspace U2=span{u1,u2}.

But I don't know how to show that span{v2,v3} contains an invariant subspace of dimension 1. I'm not sure that it's even true.

2. Feb 23, 2009

Dick

Did you prove the Schur decomposition? That any complex matrix is similar to an upper triangular matrix?

3. Feb 23, 2009

samkolb

I have this theorem:

If V is a complex vector space and T is a linear map on V, then T has an upper trianguler matrix with respect to some basis of V.

I think that this is equivalent to the Schur Theorem. I think I know how to proceed from here.

Choose a basis of V for which the matrix of T is upper triangular. Then the definition of the matrix of a linear map shows that V contains an invariant subspace of dimension j for j=1,...,dim V.

Is this right?

Thanks

4. Feb 23, 2009

Dick

Sure. The matrix form does make it pretty easy to see the invariant subspaces.