Linear Algebra Done Right · Problem 5C.2

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

A

and

B

are upper-triangular matrices of the same size, with

\alpha_1, \ldots, \alpha_n

on the diagonal of

A

and

\beta_1, \ldots, \beta_n

on the diagonal of

B

. (a) Show that

A + B

is an upper-triangular matrix with

\alpha_1 + \beta_1, \ldots, \alpha_n + \beta_n

on the diagonal. (b) Show that

AB

is an upper-triangular matrix with

\alpha_1\beta_1, \ldots, \alpha_n\beta_n

on the diagonal. The results in this exercise are used in the proof of 5.81.