360 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Lemma 13.10.7 Let D be a diagonal matrix of the form

D≡

λ 1In1 0 · · · 0

0 λ 2In2

. . ....

.... . .

. . . 00 · · · 0 λ rInr

 , (13.19)

where Ini denotes the ni×ni identity matrix and λ i ̸= λ j for i ̸= j and suppose B is a matrixwhich commutes with D. Then B is a block diagonal matrix of the form

B =

B1 0 · · · 0

0 B2. . .

......

. . .. . . 0

0 · · · 0 Br

 (13.20)

where Bi is an ni×ni matrix.

Proof: Let B=(Bi j) where Bii =Bi a block matrix as above in 13.20. Since it commuteswith D, 

B11 B12 · · · B1r

B21 B22. . . B2r

.... . . . . .

...Br1 Br2 · · · Brr



λ 1In1 0 · · · 0

0 λ 2In2

. . ....

.... . . . . . 0

0 · · · 0 λ rInr



=

λ 1In1 0 · · · 0

0 λ 2In2

. . ....

.... . . . . . 0

0 · · · 0 λ rInr



B11 B12 · · · B1r

B21 B22. . . B2r

.... . . . . .

...Br1 Br2 · · · Brr

Thus

λ jBi j = λ iBi j

Therefore, if i ̸= j,Bi j = 0. Hence B as the form which is claimed. ■

Lemma 13.10.8 Let F denote a commuting family of n×n matrices such that each A∈Fis diagonalizable. Then F is simultaneously diagonalizable.

commuting + diagonalizable ⇒ simultaneously diagonalizable

Proof: First note that if every matrix in F has only one eigenvalue, there is nothing toprove. This is because for A such a matrix,

S−1AS = λ I

and soA = λ I