Domino

⌹ — Matrix Inverse or Division — Keystroke ALT+SHIFT+= — Character 9017 or 0x2339

Note: The symbol is created by ALT+SHIFT+=; ALT+= will produce Divide (÷)
Caution: Be careful not to confuse this symbol, which is ⌹, with ⍠ which is Variant.

Usage

See also Matrix_Inverse/Divide.

Examples

⌹ Monadic Example - Matrix Inverse for a square matrix:

      Z     ⍝Variable and matrix Z has 3 rows and 3 columns of numbers, as follows:
1 2 3
0 1 4
5 6 0
      ⌹Z   ⍝The mathmatical inverse of matrix Z (domino Z) is also a 3-row by 3-col numeric matrix, as follows:
¯24  18  5
 20 ¯15 ¯4
 ¯5   4  1

⌹ Dyadic Example - Matrix Division - Least Squares Curve Fitting - Multiple Regression using Non-Square Data Matrix:

      ⍴Homes   ⍝Global variable Homes is a 23 row by 8 column matrix:
23 8
      Homes    ⍝The first row of variable/matrix Homes has nested column headers, with 22 data rows:
⍝Col#'s 1    2         3          4          5           6       7               8
 Property SqFt #Bedrooms #FullBaths #HalfBaths YrsAgoBuilt   Price Predicted Price    ⍝Note: Predicted Price col(8) is not yet calculated!
 A        2149         3          2          1          15 469.9                 0 
 B        2410         4          3          1          13 523.5                 0 
 C        2530         4          3          0          12 535                   0 
 D        2502         3          2          0          12 569.999               0 
 E        2600         5          2          1          17 588                   0 
 F        2586         4          3          1           6 589                   0 
 G        2836         3          2          1          12 599.9                 0 
 H        3300         4          2          1          10 615                   0 
 I        3000         4          3          0          11 659                   0 
 J        2723         3          2          1          12 665                   0 
 K        2800         4          3          1          18 729                   0 
 L        3700         5          3          1           4 767                   0 
 M        3900         4          3          1          19 799.9                 0 
 N        3840         6          4          1           6 945                   0 
 O        2185         3          2          0          12 515                   0 
 P        2600         4          3          0          12 579.5                 0 
 Q        4220         5          3          1          18 645.9                 0 
 R        2136         3          2          0          14 505                   0 
 S        2530         3          2          1          13 539.9                 0 
 T        2724         3          2          0          12 634.9                 0 
 U        2896         4          3          0           7 750                   0 
 V        3835         3          3          1           6 982.5                 0

      ⍝Note just below that because the first row of Homes has column headers(text) - they must be dropped from ⌹'s least-squares data fit.
      RegrCoeffs←(1↓Homes[;7])⌹1,(1 0↓Homes[;2 3 4 5 6])  ⍝Calc regr. coeff's using Home Prices(col 7, dep var Y) vs. all indep vars(X's, cols 2-6)
      ⍴RegrCoeffs     ⍝Note '⌹1,' just above means add extra coeff. for Y-intercept; plus cols(2-6)=5 indep X-type vars; RegrCoeffs has 6 elements.
6
      RegrCoeffs
234.9773005 0.1495539775 ¯41.70990009 80.27420326 16.19671155 ¯7.019567895

      Homes[1+⍳22;8]←(1,Homes[1+⍳22;2 3 4 5 6])+.×RegrCoeffs   ⍝Use inner product +.× to calc predicted prices for 22 data rows, place in col 8.

      Homes          ⍝The following is what variable Homes looks like; with Predicted Prices calculated and inserted back into matrix Homes:
 Property SqFt #Bedrooms #FullBaths #HalfBaths YrsAgoBuilt   Price Predicted Price 
 A        2149         3          2          1          15 469.9       502.6906976 
 B        2410         4          3          1          13 523.5       594.3277247 
 C        2530         4          3          0          12 535         603.0970583 
 D        2502         3          2          0          12 569.999     560.3452438 
 E        2600         5          2          1          17 588         472.6806055 
 F        2586         4          3          1           6 589         669.7862    
 G        2836         3          2          1          12 599.9       626.4929838 
 H        3300         4          2          1          10 615         668.2152651 
 I        3000         4          3          0          11 659         680.4069957 
 J        2723         3          2          1          12 665         609.5933844 
 K        2800         4          3          1          18 729         617.5559364 
 L        3700         5          3          1           4 767         808.7185666 
 M        3900         4          3          1          19 799.9       775.0457438 
 N        3840         6          4          1           6 945         854.1812909 
 O        2185         3          2          0          12 515         512.9366329 
 P        2600         4          3          0          12 579.5       613.5658367 
 Q        4220         5          3          1          18 645.9       788.2126844 
 R        2136         3          2          0          14 505         491.5693522 
 S        2530         3          2          1          13 539.9       573.7098988 
 T        2724         3          2          0          12 634.9       593.5462268 
 U        2896         4          3          0           7 750         692.9316536 
 V        3835         3          3          1           6 982.5       898.289018 

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