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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=: +/ . *         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.                      Function to be approximated
       x=: 5 %~ i. 6
       c=: (sin x) %. x ^/ i.4         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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