Matrix Divide: Difference between revisions
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{{Built-in|Matrix Divide|⌹}} is a [[dyadic function]] that performs [[wikipedia:matrix division|matrix division]] between two [[argument]]s of rank 2 or less. Some dialects automatically apply it to rank-2 [[subarray]]s of higher-rank arguments. It shares the [[glyph]] ''Quad Divide'' <source lang=apl inline>⌹</ | {{Built-in|Matrix Divide|⌹}} is a [[dyadic function]] that performs [[wikipedia:matrix division|matrix division]] between two [[argument]]s of rank 2 or less. Some dialects automatically apply it to rank-2 [[subarray]]s of higher-rank arguments. It shares the [[glyph]] ''Quad Divide'' <source lang=apl inline>⌹</syntaxhighlight> (often called ''Domino'') with the monadic function [[Matrix Inverse]]. | ||
== Examples == | == Examples == | ||
The result of <source lang=apl inline>X⌹Y</ | The result of <source lang=apl inline>X⌹Y</syntaxhighlight> is equal to <source lang=apl inline>(⌹Y)+.×X</syntaxhighlight>, which is analogous to <source lang=apl inline>X÷Y</syntaxhighlight> being equal to <source lang=apl inline>(÷Y)×X</syntaxhighlight>. As a consequence, <source lang=apl inline>X≡Y+.×X⌹Y</syntaxhighlight> is true for square matrices. | ||
<source lang=apl> | <source lang=apl> | ||
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X≡Y+.×X⌹Y | X≡Y+.×X⌹Y | ||
1 | 1 | ||
</ | </syntaxhighlight> | ||
== Applications == | == Applications == | ||
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Y⌹X | Y⌹X | ||
4.2 0.4 | 4.2 0.4 | ||
</ | </syntaxhighlight> | ||
The following example solves the linear least squares over the five points <math>(1,5), (2,1), (3,4), (4,2), (5,8)</math>. The answer is <math>y=1.9 + 0.7x</math>. | The following example solves the linear least squares over the five points <math>(1,5), (2,1), (3,4), (4,2), (5,8)</math>. The answer is <math>y=1.9 + 0.7x</math>. | ||
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Y⌹X | Y⌹X | ||
1.9 0.7 | 1.9 0.7 | ||
</ | </syntaxhighlight> | ||
When used with real vectors as both arguments, <source lang=apl inline>Y×X⌹Y</ | When used with real vectors as both arguments, <source lang=apl inline>Y×X⌹Y</syntaxhighlight> gives the [[wikipedia:Projection (linear algebra)#Finding projection with an inner product|projection]] of X onto a basis vector Y. The remaining component of X, namely <source lang=apl inline>R←X-Y×X⌹Y</syntaxhighlight>, is [[wikipedia:Orthogonality#Euclidean vector spaces|orthogonal]] to Y (<source lang=apl inline>R+.×Y</syntaxhighlight> is zero). | ||
<source lang=apl> | <source lang=apl> | ||
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⎕CT>|Y+.×X-Y×X⌹Y ⍝ ∧ is orthogonal to Y (with negligible error) | ⎕CT>|Y+.×X-Y×X⌹Y ⍝ ∧ is orthogonal to Y (with negligible error) | ||
1 | 1 | ||
</ | </syntaxhighlight> | ||
== External links == | == External links == | ||
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* [http://microapl.com/apl_help/ch_020_020_280.htm APLX] | * [http://microapl.com/apl_help/ch_020_020_280.htm APLX] | ||
* [http://wiki.nars2000.org/index.php/Matrix_Inverse/Divide NARS2000] | * [http://wiki.nars2000.org/index.php/Matrix_Inverse/Divide NARS2000] | ||
* J [https://www.jsoftware.com/help/dictionary/d131.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/percentdot#dyadic NuVoc] (as <source lang=j inline>%.</ | * J [https://www.jsoftware.com/help/dictionary/d131.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/percentdot#dyadic NuVoc] (as <source lang=j inline>%.</syntaxhighlight>) | ||
{{APL built-ins}}[[Category:Primitive functions]] | {{APL built-ins}}[[Category:Primitive functions]] | ||
Revision as of 22:09, 10 September 2022
⌹
|
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 <source lang=apl inline>⌹</syntaxhighlight> (often called Domino) with the monadic function Matrix Inverse.
Examples
The result of <source lang=apl inline>X⌹Y</syntaxhighlight> is equal to <source lang=apl inline>(⌹Y)+.×X</syntaxhighlight>, which is analogous to <source lang=apl inline>X÷Y</syntaxhighlight> being equal to <source lang=apl inline>(÷Y)×X</syntaxhighlight>. As a consequence, <source lang=apl inline>X≡Y+.×X⌹Y</syntaxhighlight> is true for square matrices.
<source lang=apl>
⎕←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 </syntaxhighlight>
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 .
<source lang=apl>
⎕←X←2 2⍴1 2 2 ¯1
1 2 2 ¯1
Y←5 8
Y⌹X
4.2 0.4 </syntaxhighlight>
The following example solves the linear least squares over the five points . The answer is .
<source lang=apl>
⎕←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 </syntaxhighlight>
When used with real vectors as both arguments, <source lang=apl inline>Y×X⌹Y</syntaxhighlight> gives the projection of X onto a basis vector Y. The remaining component of X, namely <source lang=apl inline>R←X-Y×X⌹Y</syntaxhighlight>, is orthogonal to Y (<source lang=apl inline>R+.×Y</syntaxhighlight> is zero).
<source lang=apl>
(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 </syntaxhighlight>
External links
Lesson
Documentation
- Dyalog
- APLX
- NARS2000
- J Dictionary, NuVoc (as <source lang=j inline>%.</syntaxhighlight>)