15: Linear Vector Functions

15B. Dot product as a linear vector function

15A. Dot product and vector functions S15A.

The expression +/w*x gives a sum over the argument x weighted by the vector w. For example:

   x=: 2 3 5 7 11
   w=: 0 1 2 3 4
   w*x
0 3 10 21 44
   +/w*x
78

The dot operator (.) applied to the functions +/ and * produces an equivalent function called the dot, inner, or matrix product. Thus:

   dp=: +/ . *
   w dp x
78

Moreover, dp applies to a matrix (table) left argument to produce a vector (list) of results, using successive rows of the matrix as weights. For example:

   a=: 3 1 4
   b=: 2 7 8
   c=: 2 1 0
   ]m=: a,b,:c
3 1 4
2 7 8
2 1 0

   m dp 2 3 5
29 65 7
   b dp 2 3 5
65

   g=: m with dp
   g 2 3 5
29 65 7

If m is a square matrix (as in the present example), then m with dp applies to a vector to produce a vector of the same number of elements. We will call such a function a vector function.

15B. Dot product as a linear vector function S15B.

The dot product with a vector or matrix left argument distributes over addition, and is therefore a linear function, as defined in Section 14F. For example:

   x=: 2 3 5
   y=: 0 1 2
   f=: 2 7 8 with dp

   ((f x)+(f y)) , (f x+y)
88 88
   g=: m with dp
   ((g x)+(g y)) ,: (g x+y)
38 88 8
38 88 8

The function I=: =/~ on i. yields a result called an identity matrix, because when used with dp it produces an identity function (that yields its argument unchanged). For example:

   I=: i. =/ i.
   ]i=:  I 3
1 0 0
0 1 0
0 0 1
   i with dp 2 3 5
2 3 5

A linear vector function need not be expressed as a dot product, but if it is not, the required matrix is easily determined. For example, sop=: +/\ (sums over prefixes) is a linear vector function, and the matrix m for an equivalent function m with dp may be determined as follows:

   x=: 2 3 5 7 11
   y=: 0 1 2 3 4
   sop=: +/\
   ((sop x)+(sop y)) ,: (sop x+y)
2 6 13 23 38
2 6 13 23 38

   ]i=: I # x
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1

   i dp x
2 3 5 7 11
   sop i dp x
2 5 10 17 28
   (sop i) dp x
2 5 10 17 28

   ]m=: sop i
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1

The resulting matrix provides an interesting way of looking at the sums over prefixes: the ones in the successive rows indicate the elements included in the successive sums.

15C. Matrix inverse S15C.

The inverse of a linear vector function m with dp is itself a linear vector function, and can therefore be expressed in the form mi with dp. The matrix mi may be obtained from m by applying the matrix inverse function %. . Thus:

   ]z=:  m dp x=: 2 3 5 7 11
2 5 10 17 28
   mi=: %.m
   mi dp z
2 3 5 7 11
   mi
 1  0  0  0 0
_1  1  0  0 0
 0 _1  1  0 0
 0  0 _1  1 0
 0  0  0 _1 1

The pairs _1 1 in the rows of the inverse mi indicate that the inverse of the sum over prefixes is a differencing operation, and the matrix mi may be called a difference matrix.

Exercises

Experiment with various linear vector functions such as reversal (|.), rotation (2 with |.) and 8 with A..

Comment on the matrices that represent them and their inverses.