Books/MathForTheLayman/Linear Vector Functions

15: Linear Vector Functions

15A. Dot product and vector functions

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

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.