18.4. BLOCK DIAGONAL MATRICES 435

More can be said. Recall Theorem 13.2.11 on Page 306. From this theorem, there existunitary matrices, Uk such that U∗k PkUk = Tk where Tk is an upper triangular matrix of theform 

λ k · · · ∗...

. . ....

0 · · · λ k

≡ Tk

Now let U be the block diagonal matrix defined by

U ≡

U1 · · · 0...

. . ....

0 · · · Ur

 .

By Theorem 18.4.2 there exists S such that

S−1AS =

P1 · · · 0...

. . ....

0 · · · Pr

 .

Therefore,

U∗SASU =

U∗1 · · · 0

.... . .

...0 · · · U∗r



P1 · · · 0...

. . ....

0 · · · Pr



U1 · · · 0...

. . ....

0 · · · Ur



=

U∗1 P1U1 · · · 0

.... . .

...0 · · · U∗r PrUr

=

T1 · · · 0...

. . ....

0 · · · Tr

 .

This proves most of the following corollary of Theorem 18.4.2.

Corollary 18.4.4 Let A be an n×n matrix. Then A is similar to an upper triangular, blockdiagonal matrix of the form

T ≡

T1 · · · 0...

. . ....

0 · · · Tr

where Tk is an upper triangular matrix having only λ k on the main diagonal. The diagonalblocks can be arranged in any order desired. If Tk is an mk×mk matrix, then

mk = dim(ker(A−λ kI)rk)

where the minimal polynomial of A is

p

∏k=1

(λ −λ k)rk

Furthermore, mk is the multiplicity of λ k as a zero of the characteristic polynomial of A.