. Determinant • Dot Product

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Determinant | u . v 2 _ _ | Dot Product

The phrases`-/ . *` and`+/ . *` are the determinant and permanent of square matrix arguments. More generally, the phrase`u . v` is defined in terms of a recursive expansion by minors along the first column, as discussed below. |  | For vectors and matrices, the phrase`x +/ . * y` is equivalent to the dot, inner, or matrix product of math; other rank-0 verbs such as`<.` and`*.` are treated analogously. In general,`u . v` is defined by`u@(v"(1+lv,_))`,`` restated in English below.

For example:

       x=: 1 2 3 [ m=: >1 6 4;4 1 0;6 6 8
       det=: -/ . *
       mp=: +/ . *
       x ([ ; ] ; det@] ; mp ; mp~ ; mp~@]) m
    +-----+-----+----+--------+-------+--------+
    |1 2 3|1 6 4|_112|27 26 28|25 6 42|49 36 36|
    |     |4 1 0|    |        |       | 8 25 16|
    |     |6 6 8|    |        |       |78 90 88|
    +-----+-----+----+--------+-------+--------+

The monad`u . v` is defined as illustrated below:

       DET=: 2 : 'v./@,`({."1 u. . v. $:@minors)@.(0:<{:@$) @ ,. "2'
         minors=: }."1 @ (1&([\.))

       -/ DET * m
    _112

       -/ DET * 1 16 64
    49

       -/ DET * i.3 0
    1

       +/ DET * m
    320

The definition`u@(v"(1+lv,_))` given above for the dyadic case may be re-stated in words as follows:`u` is applied to the result of`v` on lists of “left argument cells” and the right argument in toto. The number of items in a list of left argument cells must agree with the number in the right argument. Thus, if`v` has ranks`2 3` and the shapes of`x` and`y` are`2 3 4 5 6` and`4 7 8 9 10 11`,`` then there are`2 3` lists of left argument cells (each shaped`4 5 6`);`` and if the shape of a result cell is`sr`,`` the overall shape is`2 3,sr` .

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