Linear Algebra Done Right · Problem 5C.12

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

V

is finite-dimensional,

T \in \mathcal{L}(V)

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

V

, and

U

is a subspace of

V

that is invariant under

T

. (a) Prove that

T|_U

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

U

. (b) Prove that the quotient operator

T/U

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

V/U

. The quotient operator

T/U

was defined in Exercise 38 in Section 5A.