Matrix Inverse

From APL Wiki
Jump to navigation Jump to search
⌹

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

      ⎕←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


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
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)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
Retrieved from ‘https://aplwiki.com/index.php?title=Matrix_Inverse&oldid=4791’