How Do You Find Invariant Subspaces in a Complex Vector Space?

[samkolb]

Homework Statement

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.

Homework Equations

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

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.

 

[Dick]

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

 

[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

 

[Dick]

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