Linear Algebra Done Right · Problem 5C.13

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

V

is finite-dimensional and

T \in \mathcal{L}(V)

. Suppose there exists a subspace

U

V

that is invariant under

T

such that

T|_U

has an upper-triangular matrix with respect to some basis of

U

and also

T/U

has an upper-triangular matrix with respect to some basis of

V/U

. Prove that

T

has an upper-triangular matrix with respect to some basis of

V