18.4. BLOCK DIAGONAL MATRICES 433

18.4 Block Diagonal MatricesIn this section the vector space will be Cn and the linear transformations will be thosewhich result by multiplication by n×n matrices.

Definition 18.4.1 Let A and B be two n× n matrices. Then A is similar to B, written asA∼ B when there exists an invertible matrix S such that A = S−1BS.

Theorem 18.4.2 Let A be an n× n matrix. Letting λ 1,λ 2, · · · ,λ r be the distinct eigen-values of A,arranged in some order, there exist square matrices P1, · · · ,Pr such that A issimilar to the block diagonal matrix

P =

P1 · · · 0...

. . ....

0 · · · Pr

in which Pk has the single eigenvalue λ k. Denoting by rk the size of Pk it follows that rkequals the dimension of the generalized eigenspace for λ k. Furthermore, if S is the matrixsatisfying

S−1AS = P,

then S is of the form (B1 · · · Br

)where Bk =

(uk

1 · · · ukrk

)in which the columns,

{uk

1, · · · ,ukrk

}= Dk constitute a basis

for Vλ k.

Proof: By Theorem 18.3.9 and Lemma 18.3.3,

Cn =Vλ 1 ⊕·· ·⊕Vλ k

and a basis for Cn is {D1, · · · ,Dr} where Dk is a basis for Vλ k, ker(A−λ kI)rk .

LetS =

(B1 · · · Br

)where the Bi are the matrices described in the statement of the theorem. Then S−1 must beof the form

S−1 =

C1...

Cr

where CiBi = Iri×ri . Also, if i ̸= j, then CiAB j = 0 the last claim holding because A : Vλ j 7→Vλ j so the columns of AB j are linear combinations of the columns of B j and each of these