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 Matrix Inverse %.  2 _ 2 Matrix Divide

 If y is a non-singular matrix, then %.y is the inverse of y . For example: ``` mp=: +/ . * NB. Matrix product (%. ; ] ; %. mp ]) i. 2 2 +--------+---+---+ |_1.5 0.5|0 1|1 0| | 1 0|2 3|0 1| +--------+---+---+ ``` More generally, %.y is defined in terms of the dyadic case, with the left argument =i.{:\$y (an identity matrix) or, equally, by the relation (%.y)mp x ↔ x %. y . The shape of %.y is |.\$y . The vector and scalar cases are defined by using the matrix ,.y, but the shape of the result is \$y . For a non-zero vector y, the result of %.y is a vector collinear with y whose length is the reciprocal of that of y; it is called the reflection of y in the unit circle (or sphere). Thus: ``` (%. ,: ] % %.) 2 3 4 0.0689655 0.103448 0.137931 29 29 29 ``` If y is non-singular, then x%.y is (%.y) mp x . More generally, if the columns of y are linearly independent and if #x and #y agree, then x%.y minimizes the difference: ``` d=: x - y mp x %. y ``` in the sense that the magnitudes +/d*+d are minimized. Scalar and vector cases of y are treated as the one-column matrix ,.y . Geometrically, y mp x%.y is the projection of the vector x on the column space of y, the point nearest to x in the space spanned by the columns of y . Common uses of %. are in the solution of linear equations and in the approximation of functions by polynomials, as in c=: (f x)%. x ^ / i.4 .

We will illustrate the use of %. in function fitting by the sine function, showing, in particular, the maximum over the magnitudes of the differences from the function being approximated:
```   sin=: 1&o.                      NB. Function to be approximated
x=: 5 %~ i. 6
c=: (sin x) %. x ^/ i.4         NB. Use of matrix divide
,.&.>@(] ; c"_ ; sin ; c&p. ; >./@:|@(sin-c&p.)) x
+---+-----------+--------+-----------+-----------+
|  0|_5.30503e_5|       0|_5.30503e_5|0.000167992|
|0.2|    1.00384|0.198669|   0.198826|           |
|0.4|  _0.018453|0.389418|   0.389321|           |
|0.6|  _0.143922|0.564642|   0.564523|           |
|0.8|           |0.717356|   0.717524|           |
|  1|           |0.841471|   0.841416|           |
+---+-----------+--------+-----------+-----------+
```

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