| Determinant | u . v 2 _ _ | Dot Product |
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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.
|
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 .