Linear Algebra Done Right · Problem 5C.3

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

T \in \mathcal{L}(V)

is invertible and

v_1, \ldots, v_n

is a basis of

V

with respect to which the matrix of

T

is upper triangular, with

\lambda_1, \ldots, \lambda_n

on the diagonal. Show that the matrix of

T^{-1}

is also upper triangular with respect to the basis

v_1, \ldots, v_n

, with

\frac{1}{\lambda_1}, \ldots, \frac{1}{\lambda_n}

on the diagonal.