Linear Algebra

218 Chapter 3. Maps Between Spaces
Except for the lack of commutativity, matrix multiplication is algebraically
well-behaved. Below are some nice properties and more are in Exercise 23 and
Exercise 24.
2.12 Theorem If F , G, andH are matrices, and the matrix products are
defined, then the product is associative (FG)H = F (GH) and distributes over
matrix addition F (G + H) =FG+ FH and (G + H)F = GF + HF.
Proof. Associativity holds because matrix multiplication represents function
composition, which is associative: the maps (f ◦ g) ◦ h and f ◦ (g ◦ h) are equal
as both send �v to f(g(h(�v))).
Distributivity is similar. For instance, the first one goes f ◦ (g + h)(�v) =
f � (g + h)(�v) � = f � g(�v)+h(�v) � = f(g(�v)) + f(h(�v)) = f ◦ g(�v)+f ◦ h(�v) (the
third equality uses the linearity of f). QED
2.13 Remark We could alternatively prove that result by slogging through
the indices. For example, associativity goes: the i, j-th entry of (FG)H is
(fi,1g1,1 + fi,2g2,1 + ···+ fi,rgr,1)h1,j
+(fi,1g1,2 + fi,2g2,2 + ···+ fi,rgr,2)h2,j
.
+(fi,1g1,s + fi,2g2,s + ···+ fi,rgr,s)hs,j
(where F , G, andH are m×r, r×s, ands×n matrices), distribute
and regroup around the f’s
fi,1g1,1h1,j + fi,2g2,1h1,j + ···+ fi,rgr,1h1,j
+ fi,1g1,2h2,j + fi,2g2,2h2,j + ···+ fi,rgr,2h2,j
.
+ fi,1g1,shs,j + fi,2g2,shs,j + ···+ fi,rgr,shs,j
fi,1(g1,1h1,j + g1,2h2,j + ···+ g1,shs,j)
+ fi,2(g2,1h1,j + g2,2h2,j + ···+ g2,shs,j)
.
+ fi,r(gr,1h1,j + gr,2h2,j + ···+ gr,shs,j)
to get the i, j entry of F (GH).
Contrast these two ways of verifying associativity, the one in the proof and
the one just above. The argument just above is hard to understand in the sense
that, while the calculations are easy to check, the arithmetic seems unconnected
to any idea (it also essentially repeats the proof of Theorem 2.6 and so is inefficient).
The argument in the proof is shorter, clearer, and says why this property
“really” holds. This illustrates the comments made in the preamble to the chapter
on vector spaces—at least some of the time an argument from higher-level
constructs is clearer.