Exercise 5.2.17

Answers

1.
We may pick one basis α such that both [T]α and [U]α are diagonal. Let Q = [I]αβ. And we will find out that
[T]α = Q−1[T] βQ

and

[U]α = Q−1[U] βQ.
2.
Let Q be the invertible matrix who makes A and B simultaneously diagonalizable. Say β be the basis consisting of the column vectors of Q. And let α be the standard basis. Now we know that
[T]β = [I]αβ[T] α[I]βα = Q−1AQ

and

[U]β = [I]αβ[U] α[I]βα = Q−1BQ.
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2011-06-27 00:00
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