Reimagining matrices

Why is matrix multiplication defined so very differently from matrix addition? If we didn’t know these procedures, could we derive them from first principles? What might those principles be?

This post gives a simple semantic model for matrices and then uses it to systematically derive the implementations that we call matrix addition and multiplication. The development illustrates what I call “denotational design”, particularly with type class morphisms. On the way, I give a somewhat unusual formulation of matrices and accompanying definition of matrix “multiplication”.

For more details, see the linear-map-gadt source code.

Edits:

• 2012–12–17: Replaced lost $B$ entries in description of matrix addition. Thanks to Travis Cardwell.
• 2012–12018: Added note about math/browser compatibility.

Note: I’m using MathML for the math below, which appears to work well on Firefox but on neither Safari nor Chrome. I use Pandoc to generate the HTML+MathML from markdown+lhs+LaTeX. There’s probably a workaround using different Pandoc settings and requiring some tweaks to my WordPress installation. If anyone knows how (especially the WordPress end), I’d appreciate some pointers.

Matrices

For now, I’ll write matrices in the usual form: $$\left(\begin{array}{ccc}{A}_{11}& \cdots & A_{1m}\\ ⋮& \ddots & ⋮\\ A_{n1}& \cdots & A_{nm}\end{array}\right)$$

Addition

To add two matrices, we add their corresponding components. If $$A=\left(\begin{array}{ccc}{A}_{11}& \cdots & A_{1m}\\ ⋮& \ddots & ⋮\\ A_{n1}& \cdots & A_{nm}\end{array}\right)\mathrm{\text{and}}B=\left(\begin{array}{ccc}{B}_{11}& \cdots & B_{1m}\\ ⋮& \ddots & ⋮\\ B_{n1}& \cdots & B_{nm}\end{array}\right),$$ then $$A+B=\left(\begin{array}{ccc}{A}_{11}+B_{11}& \cdots & A_{1m}+B_{1m}\\ ⋮& \ddots & ⋮\\ A_{n1}+B_{n1}& \cdots & A_{nm}+B_{nm}\end{array}\right).$$ More succinctly, $$(A+B{)}_{ij}=A_{ij}+B_{ij}.$$

Multiplication

Multiplication, on the other hand, works quite differently. If $$A=\left(\begin{array}{ccc}{A}_{11}& \cdots & A_{1m}\\ ⋮& \ddots & ⋮\\ A_{n1}& \cdots & A_{nm}\end{array}\right)\mathrm{\text{and}}B=\left(\begin{array}{ccc}{B}_{11}& \cdots & B_{1p}\\ ⋮& \ddots & ⋮\\ B_{m1}& \cdots & B_{mp}\end{array}\right),$$ then $$(A\bullet B{)}_{ij}=\sum _{k=1}^m{A}_{ik}\cdot B_{kj}.$$ This time, we form the dot product of each $A$ row and $B$ column.

Why are these two matrix operations defined so differently? Perhaps these two operations are implementations of more fundamental specifications. If so, then making those specifications explicit could lead us to clear and compelling explanations of matrix addition and multiplication.

Transforming vectors

Simplifying from matrix multiplication, we have transformation of a vector by a matrix. If $$A=\left(\begin{array}{ccc}{A}_{11}& \cdots & A_{1m}\\ ⋮& \ddots & ⋮\\ A_{n1}& \cdots & A_{nm}\end{array}\right)\mathrm{\text{and}}x=\left(\begin{array}{c}{x}_1\\ ⋮\\ x_m\end{array}\right),$$ then $$A\cdot x=\left(\begin{array}{ccccc}{A}_{11}\cdot x_1& +& \cdots & +& A_{1m}\cdot x_m\\ ⋮& & \ddots & & ⋮\\ A_{n1}\cdot x_1& +& \cdots & +& A_{nm}\cdot x_m\end{array}\right)$$ More succinctly, $$(A\cdot x{)}_i=\sum _{k=1}^m{A}_{ik}\cdot x_k.$$

What’s it all about?

We can interpret matrices as transformations. Matrix addition then adds transformations:

$$(A+B)x=Ax+Bx$$

Matrix “multiplication” composes transformations:

$$(A\bullet B)x=A(Bx)$$

What kinds of transformations?

Linear transformations

Matrices represent linear transformations. To say that a transformation (or “function” or “map”) $f$ is “linear” means that $f$ preserves the structure of addition and scalar multiplication. In other words, $$\begin{array}{ccc}\hfill f(x+y)& \hfill =\hfill & fx+fy\hfill \\ \hfill f(c\cdot x)& \hfill =\hfill & c\cdot fx\hfill \end{array}$$ Equivalently, $f$ preserves all linear combinations: $$f(c_1\cdot x_1+\cdots +c_m\cdot x_m)=c_1\cdot f{x}_1+\cdots +c_m\cdot f{x}_m$$

What does it mean to say that “matrices represent linear transformations”? As we saw in the previous section, we can use a matrix to transform a vector. Our semantic function will exactly be this use, i.e., the meaning of matrix is as a function (map) from vectors to vectors. Moreover, these functions will satisfy the linearity properties above.

Representation

For simplicity, I’m going structure matrices in a unconventional way. Instead of a rectangular arrangement of numbers, use the following generalized algebraic data type (GADT):

data a ⊸ b where
  Dot   ∷ InnerSpace b ⇒
          b → (b ⊸ Scalar b)
  (:&&) ∷ VS3 a c d ⇒  -- vector spaces with same scalar field
          (a ⊸ c) → (a ⊸ d) → (a ⊸ c × d)

I’m using the notation “c × d” in place of the usual “(c,d)”. Precedences are such that “×” binds more tightly than “⊸”, which binds more tightly than “→”.

This definition builds on the VectorSpace class, with its associated Scalar type and InnerSpace subclass. Using VectorSpace is overkill for linear maps. It suffices to use modules over semirings, which means that we don’t assume multiplicative or additive inverses. The more general setting enables many more useful applications than vector spaces do, some of which I will describe in future posts.

The idea here is that a linear map results in either (a) a scalar, in which case it’s equivalent to dot v (partially applied dot product) for some v, or (b) a product, in which case it can be decomposed into two linear maps with simpler range types. Each row in a conventional matrix corresponds to Dot v for some vector v, and the stacking of rows corresponds to nested applications of (:&&).

Semantics

The semantic function, apply, interprets a representation of a linear map as a function (satisfying linearity):

apply ∷ (a ⊸ b) → (a → b)
apply (Dot b)   = dot b
apply (f :&& g) = apply f &&& apply g

where, (&&&) is from Control.Arrow.

(&&&) ∷ Arrow (↝) ⇒ (a ↝ b) → (a ↝ c) → (a ↝ (b,c))

For functions,

(f &&& g) a = (f a, g a)

Functions, linearity, and multilinearity

Functions form a vector space, with scaling and addition defined “pointwise”. Instances from the vector-space package:

instance AdditiveGroup v ⇒ AdditiveGroup (a → v) where
  zeroV   = pure   zeroV
  (^+^)   = liftA2 (^+^)
  negateV = fmap   negateV

instance VectorSpace v ⇒ VectorSpace (a → v) where
  type Scalar (a → v) = a → Scalar v
  (*^) s = fmap (s *^)

I wrote the definitions in this form to fit a template for applicative functors in general. Inlining the definitions of pure, liftA2, and fmap on functions, we get the following equivalent instances:

instance AdditiveGroup v ⇒ AdditiveGroup (a → v) where
  zeroV     = λ _ → zeroV
  f ^+^ g   = λ a → f a ^+^ g a
  negateV f = λ a → negateV (f a)

instance VectorSpace v ⇒ VectorSpace (a → v) where
  type Scalar (a → v) = a → Scalar v
  s *^ f = λ a → s *^ f a

In math, we usually say that dot product is “bilinear”, or “linear in each argument”, i.e.,

dot (s *^ u,v) ≡ s *^ dot (u,v)
dot (u ^+^ w, v) ≡ dot (u,v) ^+^ dot (w,v)

Similarly for the second argument:

dot (u,s *^ v) ≡ s *^ dot (u,v)
dot (u, v ^+^ w) ≡ dot (u,v) ^+^ dot (u,w)

Now recast the first of these properties in a curried form:

dot (s *^ u) v ≡ s *^ dot u v

i.e.,

dot (s *^ u)
 ≡ {- η-expand -}
λ v → dot (s *^ u) v
 ≡ {- "bilinearity" -}
λ v → s *^ dot u v
 ≡ {- (*^) on functions -}
λ v → (s *^ dot u) v
 ≡ {- η-contract -}
s *^ dot u

Likewise,

dot (u ^+^ v)
 ≡ {- η-expand -}
λ w → dot (u ^+^ v) w
 ≡ {- "bilinearity" -}
λ w → dot u w ^+^ dot v w
 ≡ {- (^+^) on functions -}
dot u ^+^ dot v

Thus, when “bilinearity” is recast in terms of curried functions, it becomes just linearity. (The same reasoning applies more generally to multilinearity.)

Note that we could also define function addition as follows:

f ^+^ g = add ∘ (f &&& g)

where

add = uncurry (^+^)

This uncurried form will come in handy in derivations below.

Deriving matrix operations

Addition

We’ll add two linear maps using the (^+^) operation from Data.AdditiveGroup.

(^+^) ∷ (a ⊸ b) → (a ⊸ b) → (a ⊸ b)

Following the principle of semantic type class morphisms, the specification simply says that the meaning of the sum is the sum of the meanings:

apply (f ^+^ g) ≡ apply f ^+^ apply g

which is half of the definition of “linearity” for apply.

The game plan (as always) is to use the semantic specification to derive (or “calculate”) a correct implementation of each operation. For addition, this goal means we want to come up with a definition like

f ^+^ g = <rhs>

where <rhs> is some expression in terms of f and g whose meaning is the same as the meaning as f ^+^ g, i.e., where

apply (f ^+^ g) ≡ apply <rhs>

Since Haskell has convenient pattern matching, we’ll use it for our definition of (^+^) above. Addition has two arguments, and our data type has two constructors, there are at most four different cases to consider.

First, add Dot and Dot. The specification

apply (f ^+^ g) ≡ apply f ^+^ apply g

specializes to

apply (Dot b ^+^ Dot c) ≡ apply (Dot b) ^+^ apply (Dot c)

Now simplify the right-hand side (RHS):

apply (Dot b) ^+^ apply (Dot c)
 ≡ {- apply definition -}
dot b ^+^ dot c
 ≡ {- (bi)linearity of dot, as described above -}
dot (b ^+^ c)
 ≡ {- apply definition -}
apply (Dot (b ^+^ c))

So our specialized specification becomes

apply (Dot b ^+^ Dot c) ≡ apply (Dot (b ^+^ c))

which is implied by

Dot b ^+^ Dot c ≡ Dot (b ^+^ c)

and easily satisfied by the following partial definition (replacing “≡” by “=”):

Dot b ^+^ Dot c = Dot (b ^+^ c)

Now consider the case of addition with two (:&&) constructors:

The specification specializes to

apply ((f :&& g) ^+^ (h :&& k)) ≡ apply (f :&& g) ^+^ apply (h :&& k)

As with Dot, simplify the RHS:

apply (f :&& g) ^+^ apply (h :&& k)
 ≡ {- apply definition -}
(apply f &&& apply g) ^+^ (apply h &&& apply k)
 ≡ {- See below -}
(apply f ^+^ apply h) &&& (apply g ^+^ apply k)
 ≡ {- induction -}
apply (f ^+^ h) &&& apply (g ^+^ k)
 ≡ {- apply definition -}
apply ((f ^+^ h) :&& (g ^+^ k))

I used the following property (on functions):

(f &&& g) ^+^ (h &&& k) ≡ (f ^+^ h) &&& (g ^+^ k)

Proof:

(f &&& g) ^+^ (h &&& k)
 ≡ {- η-expand -}
λ x → ((f &&& g) ^+^ (h &&& k)) x
 ≡ {- (&&&) definition for functions -}
λ x → (f x, g x) ^+^ (h x, k x)
 ≡ {- (^+^) definition for pairs -}
λ x → (f x ^+^ h x, g x ^+^ k x)
 ≡ {- (^+^) definition for functions -}
λ x → ((f ^+^ h) x, (g ^+^ k) x)
 ≡ {- (&&&) definition for functions -}
(f ^+^ h) &&& (g ^+^ k)

The specification becomes

apply ((f :&& g) ^+^ (h :&& k)) ≡ apply ((f ^+^ h) :&& (g ^+^ k))

which is easily satisfied by the following partial definition

(f :&& g) ^+^ (h :&& k) = (f ^+^ h) :&& (g ^+^ k)

The other two cases are (a) Dot and (:&&), and (b) (:&&) and Dot, but they don’t type-check (assuming that pairs are not scalars).

Composing linear maps

I’ll write linear map composition as “g ∘ f”, with type

(∘) ∷ (b ⊸ c) → (a ⊸ b) → (a ⊸ c)

This notation is thanks to a Category instance, which depends on a generalized Category class that uses the recent ConstraintKinds language extension. (See the source code.)

Following the semantic type class morphism principle again, the specification says that the meaning of the composition is the composition of the meanings:

apply (g ∘ f) ≡ apply g ∘ apply f

In the following, note that the ∘ operator binds more tightly than &&&, so f ∘ h &&& g ∘ h means (f ∘ h) &&& (g ∘ h).

Derivation

Again, since there are two constructors, we have four possible cases cases. We can handle two of these cases together, namely (:&&) and anything. The specification:

apply ((f :&& g) ∘ h) ≡ apply (f :&& g) ∘ apply h

Reasoning proceeds as above, simplifying the RHS of the constructor-specialized specification.

Simplify the RHS:

apply (f :&& g) ∘ apply h
 ≡ {- apply definition -}
(apply f &&& apply g) ∘ apply h
 ≡ {- see below -}
apply f ∘ apply h &&& apply g ∘ apply h
 ≡ {- induction -}
apply (f ∘ h) &&& apply (g ∘ h)
 ≡ {- apply definition -}
apply (f ∘ h :&& g ∘ h)

This simplification uses the following property of functions:

(p &&& q) ∘ r ≡ p ∘ r &&& q ∘ r

Sufficient definition:

(f :&& g) ∘ h = f ∘ h :&& g ∘ h

We have two more cases, specified as follows:

apply (Dot c ∘ Dot b) ≡ apply (Dot c) ∘ apply (Dot b)

apply (Dot c ∘ (f :&& g)) ≡ apply (Dot c) ∘ apply (f :&& g)

Based on types, c must be a scalar in the first case and a pair in the second. (Dot b produces a scalar, while f :&& g produces a pair.) Thus, we can write these two cases more specifically:

apply (Dot s ∘ Dot b) ≡ apply (Dot s) ∘ apply (Dot b)

apply (Dot (a,b) ∘ (f :&& g)) ≡ apply (Dot (a,b)) ∘ apply (f :&& g)

In the derivation, I won’t spell out as many details as before. Simplify the RHSs:

  apply (Dot s) ∘ apply (Dot b)
≡ dot s ∘ dot b
≡ dot (s *^ b)
≡ apply (Dot (s *^ b))

  apply (Dot (a,b)) ∘ apply (f :&& g)
≡ dot (a,b) ∘ (apply f &&& apply g)
≡ add ∘ (dot a ∘ apply f &&& dot b ∘ apply g)
≡ dot a ∘ apply f ^+^ dot b ∘ apply g
≡ apply (Dot a ∘ f ^+^ Dot b ∘ g)

I’ve used the following properties of functions:

dot (a,b)             ≡ add ∘ (dot a *** dot b)

(r *** s) ∘ (p &&& q) ≡ r ∘ p &&& s ∘ q

add ∘ (p &&& q)       ≡ p ^+^ q

apply (f ^+^ g)       ≡ apply f ^+^ apply g

Implementation:

 Dot s     ∘ Dot b     = Dot (s *^ b)
 Dot (a,b) ∘ (f :&& g) = Dot a ∘ f ^+^ Dot b ∘ g

Cross products

Another Arrow operation handy for linear maps is the parallel composition (product):

(***) ∷ (a ⊸ c) → (b ⊸ d) → (a × b ⊸ c × d)

The specification says that apply distributes over (***). In other words, the meaning of the product is the product of the meanings.

apply (f *** g) = apply f *** apply g

Where, on functions,

p *** q = λ (a,b) → (p a, q b)
        ≡ p ∘ fst &&& q ∘ snd

Simplify the specifications RHS:

  apply f *** apply g
≡ apply f ∘ fst &&& apply g ∘ snd

If we knew how to represent fst and snd via our linear map constructors, we’d be nearly done. Instead, let’s suppose we have the following functions.

compFst ∷ VS3 a b c ⇒ a ⊸ c → a × b ⊸ c
compSnd ∷ VS3 a b c ⇒ b ⊸ c → a × b ⊸ c

specified as follows:

apply (compFst f) ≡ apply f ∘ fst
apply (compSnd g) ≡ apply g ∘ snd

With these two functions (to be defined) in hand, let’s try again.

  apply f *** apply g
≡ apply f ∘ fst &&& apply g ∘ snd
≡ apply (compFst f) &&& apply (compSnd g)
≡ apply (compFst f :&& compSnd g)

Composing with fst and snd

I’ll elide even more of the derivation this time, focusing reasoning on the meanings. Relating to the representation is left as an exercise. The key steps in the derivation:

dot a     ∘ fst ≡ dot (a,0)

(f &&& g) ∘ fst ≡ f ∘ fst &&& g ∘ fst

dot b     ∘ snd ≡ dot (0,b)

(f &&& g) ∘ snd ≡ f ∘ snd &&& g ∘ snd

Implementation:

compFst (Dot a)   = Dot (a,zeroV)
compFst (f :&& g) = compFst f &&& compFst g

compSnd (Dot b)   = Dot (zeroV,b)
compSnd (f :&& g) = compSnd f &&& compSnd g

where zeroV is the zero vector.

Given compFst and compSnd, we can implement fst and snd as linear maps simply as compFst id and compSnd id, where id is the (polymorphic) identity linear map.

Reflections

This post reflects an approach to programming that I apply wherever I’m able. As a summary:

• Look for an elegant what behind a familiar how.
• Define a semantic function for each data type.
• Derive a correct implementation from the semantics.

You can find more examples of this methodology elsewhere in this blog and in the paper Denotational design with type class morphisms.