Matrix multiplication and dot-product

Question

Exercise 2.37. Suppose we represent vectors v = (vi) as sequences of numbers, and matrices m = (mij) as sequences of vectors (the rows of the matrix). For example, the matrix

is represented as the sequence ((1 2 3 4) (4 5 6 6) (6 7 8 9)). With this representation, we can use sequence operations to concisely express the basic matrix and vector operations. These operations (which are described in any book on matrix algebra) are the following:

We can define the dot product as17

(define (dot-product v w)
(accumulate + 0 (map * v w)))

Fill in the missing expressions in the following procedures for computing the other matrix operations. (The procedure accumulate-n is defined in exercise 2.36.)

(define (matrix-*-vector m v)
  (map <??> m))
(define (transpose mat)
  (accumulate-n <??> <??> mat))
(define (matrix-*-matrix m n)
  (let ((cols (transpose n)))
    (map <??> m)))

I wrote the following solutions:

(define (accumulate op initial items)
  (if (null? items)
      initial
      (op (car items) (accumulate op initial (cdr items)))))

(define (accumulate-n op init seqs)
  (if (null? (car seqs))
      null
      (cons (accumulate op init (map (lambda (x) (car x)) seqs))
            (accumulate-n op init (map (lambda (x) (cdr x)) seqs)))))

(define (dot-product v w)
  (accumulate + 0 (map * v w)))

(define (matrix-*-vector m v)
  (accumulate-n +
                0
                (map (lambda (row) 
                       (map (lambda (i j) (* i j)) row v)) m)))

(define (transpose mat)
  (accumulate-n (lambda (x y) (cons x y)) null mat))

(define (matrix-*-matrix m n) 
  (let ((cols (transpose n)))
    (map (lambda (m-row) 
           (accumulate-n +
                         0
                         (map (lambda (col) 
                                (map * m-row col))
                              cols))) m)))


(define m1 (list (list 1 2 3 4) (list 4 5 6 6) (list 6 7 8 9)))
(define m2 (list (list 10 20 30) (list 40 50 60) (list 70 80 90) (list 100 110 120)))

(define v1 (list 10 20 30 40))

What do you think?

Answer 1

Your definition of transpose is correct, although it can be written succinctly as:

(define (transpose mat)
  (accumulate-n cons null mat))

The remaining two definitions may be written more succinctly by using functions defined earlier. Notice that multiplying a matrix to a vector is conceptually the same as obtaining the dot-product of the vector and each row of the matrix. Thus, one may define matrix-*-vector as:

(define (matrix-*-vector m v)
  (map (lambda (row) (dot-product v row)) m))

The above definition satisfies the requirement of the exercise.

Similarly, matrix multiplication of A to B is the same as transposing B and performing a matrix-vector multiplication with each row of A. Thus, one may define matrix-*-matrix as:

(define (matrix-*-matrix m n)
  (let
      ((cols (transpose n)))
    (map (lambda (row) (matrix-*-vector cols row)) m)))

If you wish to go farther than the exercise requires, you may consider using function currying. Currying allows one to apply a function to arguments incrementally. If your implementation of Scheme provides the curry function, you may write:

(define (matrix-*-vector m v)
  (map (curry dot-product v) m))

(define (matrix-*-matrix m n)
  (map (curry matrix-*-vector (transpose n)) m))