I came across this problem one of my practice worksheets and I was stumped as to how I would go about solving this.
Let $T : V \rightarrow V$ be a linear operator on a finite dimensional vector space $V$ over $\Bbb C$. Assume that $T$ has the following property: for each invariant subspace $U \subset V$ , there exists an invariant subspace $W \subset V$ such that $V = U \oplus W$. Show that $T$ has a diagonal matrix with respect to some basis of $V$.
One thing I know is that as a direct result of the last assumption, that we have $\dim V$ linearly independent vectors that make up the bases of $V$ and $U$, but I'm not sure how exactly to go from here to showing that these vectors are eigenvectors of T.
Any help would be greatly appreciated. Thanks!