Exercise 5.2.18

Answers

1.

Let

β

be the basis makes

T

and

U

simultaneously diagonalizable. We know that each pair of diagonal matrices commute. So we have

{[T]}_{β} {[U]}_{β} = {[U]}_{β} {[T]}_{β} .

And this means $T$ and $U$ commute.

2.

Let

Q

be the invertible matrix who makes

A

and

B

simultaneously diagonalizable. Thus we have

(Q^{- 1} AQ) (Q^{- 1} BQ) = (Q^{- 1} BQ) (Q^{- 1} AQ) .

And this means that $A$ and $B$ commute since $Q$ is invertible.

jephianlin

2011-06-27 00:00