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 ⌹ (often called Domino) with the dyadic function Matrix Divide. These functions were added to APL\360 in 1970[1] and are widely supported in modern APL.

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. 

      ⎕←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     
 ∧

When the argument is a scalar or vector, or the given matrix has more rows than columns (r>c where r c≡⍴X), 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 (+X)÷X+.×+X. For a non-square matrix, the result equals (+⍉X)⌹(+⍉X)+.×X (where +⍉X is the conjugate transpose of X). 

      (⌹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

External links

Lesson

Documentation

References

  1. ↑ "Report of the APL SHARE conference" (pdf). APL Quote-Quad Volume 2, Number 3. 1970-09.
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