Chapter 12

Self Adjoint Operators

12.1 Simultaneous Diagonalization

Recall the following definition of what it means for a matrix to be diagonalizable.

Definition 12.1.1 Let A be an n× n matrix. It is said to be diagonalizable if there existsan invertible matrix S such that

S−1AS = D

where D is a diagonal matrix.

Also, here is a useful observation.

Observation 12.1.2 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 = λksk

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

C ≡

(A 0

0 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

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

(SA 0

0 SB

)is such that

S−1CS = DC , a diagonal matrix.Conversely, suppose C is diagonalized by S = (s1, · · · , sn+m) . Thus S has columns si.

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 (λ1, · · · , λn+m) =

(D1 0

0 D2

)

295