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.


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

      X2 21 2 3 4
1 2
3 4
      Y2 25 6 7 8
5 6
7 8
 5  4
¯4 ¯3
 5  4
¯4 ¯3


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 .

      X2 21 2 2 ¯1
1  2
2 ¯1
      Y5 8
4.2 0.4

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

1 1
1 2
1 3
1 4
1 5
      Y5 1 4 2 8
1.9 0.7

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

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

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