Matrix Inverse

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⌹

Matrix Inverse (⌹) is a monadic primitive function that returns the inverse of a simple numeric array of rank 2 or lower. 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 dyadic function Matrix Divide.

Examples

Matrix Inverse computes the ordinary inverse if the argument is a square matrix. DOMAIN ERROR is raised if the given matrix is not invertible.

<source lang=apl> 

     ⎕←M←2 2⍴3 4 4 5

3 4 4 5 

     ⎕←R←⌹M

¯5 4 

4 ¯3
     R+.×M

1 0 0 1 

     ⌹2 2⍴0

DOMAIN ERROR 

⌹2 2⍴0     
∧

</syntaxhighlight>

When the argument is a scalar or vector, or the given matrix has more rows than columns (<source lang=apl inline>r>c</syntaxhighlight> where <source lang=apl inline>r c≡⍴X</syntaxhighlight>), Matrix Inverse computes specific forms of generalized inverse called Moore-Penrose inverse. For a scalar, the result is the reciprocal of the argument; for a vector, the result equals <source lang=apl inline>(+X)÷X+.×+X</syntaxhighlight>. For a non-square matrix, the result equals <source lang=apl inline>(+⍉X)⌹(+⍉X)+.×X</syntaxhighlight> (where <source lang=apl inline>+⍉X</syntaxhighlight> is the conjugate transpose of X).

<source lang=apl> 

     (⌹2)(⌹2J1)

0.5 0.4J¯0.2 

     ÷2 2J1

0.5 0.4J¯0.2 

     (⌹3 1)(⌹2 1 1J2)

┌───────┬────────────────┐ │0.3 0.1│0.2 0.1 0.1J¯0.2│ └───────┴────────────────┘ 

     {(+⍵)÷⍵+.×+⍵}¨ (3 1) (2 1 1J2)

┌───────┬────────────────┐ │0.3 0.1│0.2 0.1 0.1J¯0.2│ └───────┴────────────────┘ 

     (⌹3 1)(⌹2 1 1J2) +.ר (3 1)(2 1 1J2)

1 1 

     ⎕←M←3 2⍴1 ¯1 0J1 1 ¯1 0J1
1   ¯1  
0J1  1

¯1 0J1 

     ⎕←R←⌹M
0.5J¯0.5 0.25J¯0.25 ¯0.25J¯0.25

¯0.5J¯0.5 0.25J¯0.25 ¯0.25J¯0.25 

     R≡{(+⍉⍵)⌹(+⍉⍵)+.×⍵} M

1 

     R+.×M
1.0000E000J¯5.5511E¯17 0

¯2.7756E¯17J05.5511E¯17 1 </syntaxhighlight>

External links

Lesson

Documentation


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
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Non-Scalar
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Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
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Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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