Matrix Divide
⌹

Matrix Divide (⌹
) is a dyadic function that performs matrix division between two arguments of rank 2 or less. Some dialects automatically apply it to rank2 subarrays of higherrank 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 MoorePenrose 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←XY×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 XY×X⌹Y ⍝ The remaining component in X ¯1.9 5.7 ⎕CT>Y+.×XY×X⌹Y ⍝ ∧ is orthogonal to Y (with negligible error) 1
External links
Lesson
Documentation
 Dyalog
 APLX
 NARS2000
 J Dictionary, NuVoc (as
%.
)
References
 ↑ "Report of the APL SHARE conference" (pdf). APL QuoteQuad Volume 2, Number 3. 197009.