358 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Also, here is a useful observation.

Observation 13.10.2 If A is an n× n matrix and AS = SD for D a diagonal matrix, theneach column of S is an eigenvector or else it is the zero vector. This follows from observingthat for sk the kth column of S and from the way we multiply matrices,

Ask = λ ksk

It is sometimes interesting to consider the problem of finding a single similarity trans-formation which will diagonalize all the matrices in some set.

Lemma 13.10.3 Let A be an n×n matrix and let B be an m×m matrix. Denote by C thematrix

C ≡

(A 00 B

).

Then C is diagonalizable if and only if both A and B are diagonalizable.

Proof: Suppose S−1A ASA = DA and S−1

B BSB = DB where DA and DB are diagonal ma-

trices. You should use block multiplication to verify that S ≡

(SA 00 SB

)is such that

S−1CS = DC, a diagonal matrix.Consider the converse that C is diagonalizable. It is necessary to show that A has a basis

of eigenvectors for Fn and that B has a basis of eigenvectors in Fm. Thus S has columns si.

Suppose C is diagonalized by S =(

s1 · · · sn+m

). For each of these columns, write

in the form

si =

(xi

yi

)where xi ∈ Fn and where yi ∈ Fm. The result is

S =

(S11 S12

S21 S22

)

where S11 is an n×n matrix and S22 is an m×m matrix. Then there is a diagonal matrix,D1 being n×n and D2 m×m such that

D = diag(λ 1, · · · ,λ n+m) =

(D1 00 D2

)

such that (A 00 B

)(S11 S12

S21 S22

)

=

(S11 S12

S21 S22

)(D1 00 D2

)