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.

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.

      M2 23 4 4 5
3 4
4 5
      RM
¯5  4
 4 ¯3
      R+.×M
1 0
0 1
      2 20
DOMAIN ERROR
 2 20     
 

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 0.10.2 0.1 0.1J¯0.2
└───────┴────────────────┘
      {(+)÷+.×+}¨ (3 1) (2 1 1J2)
┌───────┬────────────────┐
0.3 0.10.2 0.1 0.1J¯0.2
└───────┴────────────────┘
      (3 1)(2 1 1J2) +.ר (3 1)(2 1 1J2)
1 1

      M3 21 ¯1 0J1 1 ¯1 0J1
 1   ¯1  
 0J1  1  
¯1    0J1
      RM
 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


APL built-ins [edit]
Primitive functions
Scalar
Monadic ConjugateNegateSignumReciprocalMagnitudeExponentialNatural LogarithmFloorCeilingFactorialNotPi TimesRollTypeImaginarySquare Root
Dyadic AddSubtractTimesDivideResiduePowerLogarithmMinimumMaximumBinomialComparison functionsBoolean functions (And, Or, Nand, Nor) ∙ GCDLCMCircularComplexRoot
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