Difference between revisions of "Matrix Inverse/Divide"
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Latest revision as of 09:28, 16 October 2019
Caution: Be careful not to confuse this symbol, which is ⌹, with ⍠ which is Variant.


R is a numeric scalar, vector or matrix; otherwise signal a RANK ERROR.  
Z is a numeric array of rank ⍴⍴R, and shape ⌽⍴R. 
This feature implements matrix inversion on Boolean, integer, or floating point arguments using Singular Value Decomposition. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.
Matrix inverse (⌹R) and matrix division (L⌹R) on Rational, VFP, or Ball arguments each have two limitations above and beyond that of normal conformability:
for a square (=/⍴R) right argument that it be nonsingular, and
for an overdetermined (>/⍴R) or underdetermined (</⍴R) right argument that the symmetric matrix (⍉R)+.×R be nonsingular.
These limitations are due to the algorithm (GaussJordan Elimination) used to implement Matrix Inverse/Divide on Rational and VFP numbers.
Overdetermined matrices are evaluated equivalently to the expression (⌹(⍉R)+.×R)+.×⍉R.
Underdetermined matrices are evaluated equivalently to the expression (⍉R)+.×⌹R+.×⍉R.
For example,
⌹3 3⍴0 0 0 0 0 0 0 0 0 0 ⌹3 3⍴1 2 3 4 ¯0.1944444444 0.2777777778 0.02777777778 0.05555555556 ¯0.2222222222 0.2777777778 0.3611111111 0.05555555556 ¯0.1944444444 ⌹3 3⍴1 2 3 4x ¯7r36 5r18 1r36 1r18 ¯2r9 5r18 13r36 1r18 ¯7r36


L is a numeric scalar, vector or matrix; otherwise signal a RANK ERROR.  
R is a numeric scalar, vector or matrix; otherwise signal a RANK ERROR.  
If either L or R is a scalar or vector, it is coerced to a matrix by (L R)←⍪¨L R. After this coercion, if the two matrices have a different number of rows, signal a LENGTH ERROR.  
Z is a numeric array. Before the above coercion of L and R, the rank of Z is ¯2+2⌈(⍴⍴L)+⍴⍴R, and the shape is (1↓⍴R),1↓⍴L. 
This feature implements matrix division on Boolean, integer and floating point arguments using Singular Value Decomposition. In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.
As noted above, Matrix Division on Rational, VFP, or Ball arguments uses a different algorithm and has slightly more restrictive conformability requirements.
for a square (=/⍴R) right argument that it be nonsingular, and
for an overdetermined (>/⍴R) or underdetermined (</⍴R) right argument that the symmetric matrix (⍉R)+.×R be nonsingular.
Overdetermined matrices are evaluated equivalently to the expression (⌹(⍉R)+.×R)+.×(⍉R)+.×L.
Underdetermined matrices are evaluated equivalently to the expression (⍉R)+.×⌹R+.×(⍉R)+.×L.
For example,
a←3 3⍴0 a⌹a 0 0 0 0 0 0 0 0 0 1 2 3⌹3 3⍴1 2 3 4 0.4444444444 0.4444444444 ¯0.1111111111 1 2 3⌹3 3⍴1 2 3 4x 4r9 4r9 ¯1r9
For several more everyday problemsolving examples using Matrix Inverse see also Domino Symbol ⌹.
See Also  
System Commands  System Variables and Functions  Operators 
Keyboard  
A+S  ⍪  ≡  ≢  ⍒  ⍋  ⌽  ⍉  ⊖  ⍟  ⍱  ⍲  ⍠  ⌹  
Alt  ⋄  ¨  ¯  <  ≤  ∅  ≥  >  ≠  ∨  ∧  ×  ÷  
Sh  ~  !  @  #  $  %  ^  &  *  (  )  _  +  
Key  `  1  2  3  4  5  6  7  8  9  0    =  
A+S  ⍷  √  ⍨  ⍸  ⍥  ⍣  ⍞  ⍬  ⊣  
Alt  ?  ⍵  ∊  ⍴  ∼  ↑  ↓  ⍳  ○  π  ←  →  ⊢  
Sh  Q  W  E  R  T  Y  U  I  O  P  {  }    
Key  q  w  e  r  t  y  u  i  o  p  [  ]  \  
A+S  §  ⍢  ⍙  ⍤  ⌷  
Alt  ⍺  ⌈  ⌊  ∞  ∇  ∆  ∘  ‼  ⎕  ⍎  ⍕  
Sh  A  S  D  F  G  H  J  K  L  :  "  
Key  a  s  d  f  g  h  j  k  l  ;  '  
A+S  ⊆  ⊇  ⍡  ⍭  ⊙  
Alt  ⊂  ⊃  ∩  ∪  ⊥  ⊤  ⍦  ⍝  ⍀  ⌿  
Sh  Z  X  C  V  B  N  M  <  >  ?  
Key  z  x  c  v  b  n  m  ,  .  / 
NARS 2000 Lang Tool Bar 
←  →  +    ×  ÷  *  ⍟  ⌹  ○  !  ?  √    ⌈  ⌊  ⊥  ⊤  ⊣  ⊢  
≡  ≢  <  ≤  =  ≥  >  ≠  ∨  ∧  ⍱  ⍲  ↑  ↓  ⊂  ⊃  ⌷  ⍋  ⍒  
⍳  ∊  ⍸  ⍷  ∪  ∩  ⊆  ⊇  ~  §  π  ..  ,  ⍪  ⍴  ⌽  ⊖  ⍉  
/  \  ⌿  ⍀  ⊙  ¨  ⍨  ⍤  ⍣  ⍡  ⍥  ⍦  ⍥  .  ∘  ⍠  ‼  ⌻  ∂  ⍞  ⎕  ⍎  ⍕  
⋄  ⍝  ∇  ∆  ⍙  _  ⍺  ⍵  ¯  ⍬  ∞  ∅  
Second Row  i j k  i j k l  g  p  r  v  x 