Matrix Divide R←X⌹Y

Y must be a simple numeric array of rank 2 or less. X must be a simple numeric array of rank 2 or less. Y must be non-singular. A scalar argument is treated as a matrix with one-element. If Y is a vector, it is treated as a single column matrix. If X is a vector, it is treated as a single column matrix. The number of rows in X and Y must be the same. Y must have at least the same number of rows as columns.

R is the result of matrix division of X by Y. That is, the matrix product Y+.×R is X.

R is determined such that (X-Y+.×R)*2 is minimised.

The shape of R is (1↓⍴Y),1↓⍴X.

Examples

      ⎕PP←5

      B
3 1 4
1 5 9
2 6 5

      35 89 79 ⌹ B
2.1444 8.2111 5.0889

      A
35 36
89 88
79 75

      A ⌹ B
2.1444 2.1889
8.2111 7.1222
5.0889 5.5778

If there are more rows than columns in the right argument, the least squares solution results. In the following example, the constants a and b which provide the best fit for the set of equations represented by P = a + bQ are determined:

      Q
1 1
1 2
1 3
1 4
1 5
1 6

      P
12.03 8.78 6.01 3.75 ¯0.31 ¯2.79

      P⌹Q
14.941 ¯2.9609

Example: linear regression on complex numbers

      x←j⌿¯50+?2 13 4⍴100
      y←(x+.×3 4 5 6) + j⌿0.0001ׯ50+?2 13⍴100
      ⍴x
13 4
      ⍴y
13
      y ⌹ x
3J0.000011066 4J¯0.000018499 5J0.000005745 6J0.000050328
      ⍝ that is, y⌹x recovered the coefficients 3 4 5 6

Additional Information

      x⌹y ←→ (⌹(⍉y)+.×y)+.×(⍉y)+.×x

(Use +⍉ instead of ⍉ for complex y.)

This equivalence, familiar to mathematicians and statisticians, explains

• the conformability requirements for ⌹
• how to compute the result for tall matrices from the better known square matrix case