Linear Algebra Done Right · Problem 22

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · block-diagonal-matrices

Problem

Suppose

A

and

B

are block diagonal matrices of the form

A = \begin{pmatrix} A_1 & & 0 \\ & \ddots & \\ 0 & & A_m \end{pmatrix}, \quad B = \begin{pmatrix} B_1 & & 0 \\ & \ddots & \\ 0 & & B_m \end{pmatrix},

where

A_k

and

B_k

are square matrices of the same size for each

k = 1, \ldots, m

. Show that

AB

is a block diagonal matrix of the form

AB = \begin{pmatrix} A_1 B_1 & & 0 \\ & \ddots & \\ 0 & & A_m B_m \end{pmatrix}.