# 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 rank-2 subarrays of higher-rank arguments. It shares the glyph Quad Divide (often called Domino) with the monadic function Matrix Inverse.

## Examples

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
XY
5  4
¯4 ¯3
(Y)+.×X
5  4
¯4 ¯3
XY+.×XY
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 ${\displaystyle x+2y=5,2x-y=8}$. The answer is ${\displaystyle x=4.2,y=0.4}$.

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

The following example solves the linear least squares over the five points ${\displaystyle (1,5),(2,1),(3,4),(4,2),(5,8)}$. The answer is ${\displaystyle y=1.9+0.7x}$.

X1,⍪⍳5
1 1
1 2
1 3
1 4
1 5
Y5 1 4 2 8
YX
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)
XY
1.3
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)
1