How can I find invariant subspaces of a particular matrix A=$\begin{pmatrix}1&3 \\ 1 &-1\end{pmatrix}$ without using any concepts of eigenvalues and eigenvectors?
I've already found that {0}, and $\mathbb{R}^2$ are invariant subspaces. But I have no clue how to go about finding the invariant subspaces with dimension 1.
Best Answer
You need to solve for a non-zero $V$ in $$AV=\lambda V$$
The same idea of finding eigen-vectors. I found two vectors, $$\binom {1}{1/3}$$ and $$\binom {1}{-1}$$
So the span of each vector gives you a one dimensional invariant subspace.