Matrix Divide

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⌹

Matrix Divide (⌹) is a dyadic function that performs matrix division between two arguments of rank 2 or less. Some dialects automatically apply it to rank-2 subarrays of higher-rank arguments. It shares the glyph Quad Divide ⌹ (often called Domino) with the monadic function Matrix Inverse. These functions were added to APL\360 in 1970[1] and are widely supported in modern APL.

Examples

The result of X⌹Y is equal to (⌹Y)+.×X, which is analogous to X÷Y being equal to (÷Y)×X. As a consequence, X≡Y+.×X⌹Y is true for square matrices. 

      ⎕←X←2 2⍴1 2 3 4
1 2
3 4
      ⎕←Y←2 2⍴5 6 7 8
5 6
7 8
      X⌹Y
 5  4
¯4 ¯3
      (⌹Y)+.×X
 5  4
¯4 ¯3
      X≡Y+.×X⌹Y
1

Applications

From the properties of Moore-Penrose inverse (which Matrix Inverse uses), Matrix Divide can not only be used to solve a system of linear equations, but also to find the linear least squares solution to an overdetermined system.

The following example solves the system of equations . The answer is . 

      ⎕←X←2 2⍴1 2 2 ¯1
1  2
2 ¯1
      Y←5 8
      Y⌹X
4.2 0.4

The following example solves the linear least squares over the five points . The answer is . 

      ⎕←X←1,⍪⍳5
1 1
1 2
1 3
1 4
1 5
      Y←5 1 4 2 8
      Y⌹X
1.9 0.7

When used with real vectors as both arguments, Y×X⌹Y gives the projection of X onto a basis vector Y. The remaining component of X, namely R←X-Y×X⌹Y, is orthogonal to Y (R+.×Y is zero). 

      (X Y)←(2 7)(3 1)
      X⌹Y
1.3
      Y×X⌹Y  ⍝ Projection of X onto Y
3.9 1.3
      X-Y×X⌹Y  ⍝ The remaining component in X
¯1.9 5.7
      ⎕CT>|Y+.×X-Y×X⌹Y  ⍝ ∧ is orthogonal to Y (with negligible error)
1

External links

Lesson

Documentation

References

  1. ↑ "Report of the APL SHARE conference" (pdf). APL Quote-Quad Volume 2, Number 3. 1970-09.
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