126 CHAPTER 6. DIRECT SUMS AND BLOCK DIAGONAL MATRICES

Lemma 6.1.5 Let Li be in L (V,V ) and suppose for i ̸= j,LiL j = L jLi and also Li is oneto one on ker(L j) whenever i ̸= j. Then

ker

(p

∏i=1

Li

)= ker(L1)⊕+ · · ·+⊕ker(Lp)

Here ∏pi=1 Li is the product of all the linear transformations. It signifies

Lp ◦Lp−1 ◦ · · · ◦L1

or the product in any other order since the transformations commute.

Proof : Note that since the operators commute, L j : ker(Li)→ ker(Li). Here is why.If Liy = 0 so that y ∈ ker(Li) , then LiL jy = L jLiy = L j0 = 0 and so L j : ker(Li) 7→ker(Li). Next observe that it is obvious that, since the operators commute, ∑

pi=1 ker(Lp)⊆

ker(∏

pi=1 Li

).

Next, why is ∑i ker(Lp) = ker(L1)⊕·· ·⊕ker(Lp)? Suppose ∑pi=1 vi = 0, vi ∈ ker(Li) ,

but some vi ̸= 0. Then do ∏ j ̸=i L j to both sides. Since the linear transformations commute,this results in (

∏j ̸=i

L j

)(vi) = 0

which contradicts the assumption that these L j are one to one on ker(Li) and the observationthat they map ker(Li) to ker(Li). Thus if ∑i vi = 0, vi ∈ ker(Li) then each vi = 0. It followsthat

ker(L1)⊕+ · · ·+⊕ker(Lp)⊆ ker

(p

∏i=1

Li

)(*)

From Sylvester’s theorem and the observation about direct sums in Lemma 6.0.3,

p

∑i=1

dim(ker(Li)) = dim(ker(L1)⊕+ · · ·+⊕ker(Lp))

≤ dim

(ker

(p

∏i=1

Li

))≤

p

∑i=1

dim(ker(Li))

which implies all these are equal. Now in general, if W is a subspace of V, a finite dimen-sional vector space and the two have the same dimension, then W = V, Lemma 6.1.2. Itfollows from * that

ker(L1)⊕+ · · ·+⊕ker(Lp) = ker

(p

∏i=1

Li

)■

So how does the above situation occur? First recall the following theorem and corollaryabout polynomials. It was Theorem 6.1.6 and Corollary 6.1.7 proved earlier.

Theorem 6.1.6 Let f (λ ) be a nonconstant polynomial with coefficients in F. Then thereis some a ∈ F such that f (λ ) = a∏

ni=1 φ i (λ ) where φ i (λ ) is an irreducible nonconstant

monic polynomial and repeats are allowed. Furthermore, this factorization is unique in thesense that any two of these factorizations have the same nonconstant factors in the product,possibly in different order and the same constant a.