Linear Algebra

Section IV. Matrix Operations 213Except for the lack of commutativity, matrix multiplication is algebraicallywell-behaved. Below are some nice properties and more are in Exercise 23 andExercise 24.2.12 Theorem If F , G, and H are matrices, and the matrix products aredefined, then the product is associative (F G)H = F (GH) and distributes overmatrix addition F (G + H) = F G + F H and (G + H)F = GF + HF .Proof. Associativity holds because matrix multiplication represents functioncomposition, which is associative: the maps (f ◦ g) ◦ h and f ◦ (g ◦ h) are equalas 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) (thethird equality uses the linearity of f).QED2.13 Remark We could alternatively prove that result by slogging throughthe indices. For example, associativity goes: the i, j-th entry of (F G)H is(f i,1 g 1,1 + f i,2 g 2,1 + · · · + f i,r g r,1 )h 1,j+ (f i,1 g 1,2 + f i,2 g 2,2 + · · · + f i,r g r,2 )h 2,j.+ (f i,1 g 1,s + f i,2 g 2,s + · · · + f i,r g r,s )h s,j(where F , G, and H are m×r, r×s, and s×n matrices), distributef i,1 g 1,1 h 1,j + f i,2 g 2,1 h 1,j + · · · + f i,r g r,1 h 1,j+ f i,1 g 1,2 h 2,j + f i,2 g 2,2 h 2,j + · · · + f i,r g r,2 h 2,j..and regroup around the f’s+ f i,1 g 1,s h s,j + f i,2 g 2,s h s,j + · · · + f i,r g r,s h s,jf i,1 (g 1,1 h 1,j + g 1,2 h 2,j + · · · + g 1,s h s,j )+ f i,2 (g 2,1 h 1,j + g 2,2 h 2,j + · · · + g 2,s h s,j ).+ f i,r (g r,1 h 1,j + g r,2 h 2,j + · · · + g r,s h s,j )to get the i, j entry of F (GH).Contrast these two ways of verifying associativity, the one in the proof andthe one just above. The argument just above is hard to understand in the sensethat, while the calculations are easy to check, the arithmetic seems unconnectedto 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 chapteron vector spaces — at least some of the time an argument from higher-levelconstructs is clearer.