# Vocabulary/dot

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`u . v y`Determinant Conjunction

Rank 2 *-- operates on tables of y --*
WHY IS THIS IMPORTANT?

`-/ . * y` is the *determinant* of square matrix `y` .

`+/ . * y` is thepermanent of square matrix `y` .

]y=: 3 3 $ 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 -/ . * y 1

### More Information

1. `-/ . * y`, which calculates the standard determinant, is fast.

1. `u . v y ` is defined for general `u` and `v` in terms of expansion by minors and cofactors. Minors are calculated in the usual way and dyadic `u . v` is applied between the minors and cofactors. This is prohibitively slow for arrays larger than, say, 15x15.

1. The space before `.` is required.

`x u . v y`Matrix Product Conjunction

Rank Infinity *-- operates on x and y as a whole --*
WHY IS THIS IMPORTANT?

`x +/ . * y` is the *inner product* or *matrix product* of matrices `x` and `y`.

]y=: i.2 3 0 1 2 3 4 5 x=: 0 1 x +/ . * y 3 4 5

### Common Uses

1. Multiply matrices `x` and `y`

]x=: 3 3 $ 0 0 1 1 0 0 0 1 0 NB. permutation mtx (unitary) 0 0 1 1 0 0 0 1 0 ]y=: |:x NB. Transpose of x is its inverse 0 1 0 0 0 1 1 0 0 x +/ . * y NB. so the matrix product should be the Identity matrix 1 0 0 0 1 0 0 0 1

2. Transitive closure of a connection matrix (If A connects to B and B connects to C, make A connect to C)

tc=: (+. +./ .*.~)^:_ NB. Compute transitive closure ]cmtx =. 4 4 $ 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 NB. A connection matrix 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 tc cmtx NB. closure 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0

### More Information

1. `x +/ . * y` is the standard matrix product of `x` and `y`, also called the *inner product*.

2. Alternative inner products can be defined by using `x u/ . v y` where `v` is a verb with rank 0. When such a verb is applied to `x` and `y` arguments that are tables, the result `r` will be a table where each atom of `r` is an inner product of one row of `x` and one column of `y`; formally, `(<i,j){r` in this special case is `u/ (i { x) v (j {"1 y)`.

3. `x u . v y` is also defined for other cases, where the arguments are not tables or `u` and `v` have different forms or ranks, as `x u@(v"(1+lv,_)) y`, where `lv` is the left rank of dyadic `v`. This means that the operand `x` is broken up into pieces that have rank one higher than `lv`; i.e. `x` is broken up into an **array of lists of left-argument cells to v**. Then, for each such list-of-cells from

`x`,

`v`is applied

**between each**, which creates a list of result cells that are assembled into an array result for the list-of-

`x`-cell and the entire`y``x`-cells; then

`u`is applied to that array to produce the part of the result of

`u . v`resulting from that list-of-

`x`-cells. The results of

`u`on all the lists-of-

`x`-cells are assembled into the final result.

To see how this applies, consider the usual case of `+/ . *` applied to matrices. The rank of dyad `*` is `0 0`, so `lv` is `0`. Therefore the verb executes as `+/@(*"1 _)`. The pieces of `x` referred to above are then the 1-cells of `x`, that is, the rows. Each row is processed as `row * y` which multiplies each entire row of `y` by the corresponding atom of `x`. Then `+/` is applied, which sums the rows to produce one row of the result.

4. The space before `.` is required.

5. Some walkthrough explanations

6. As a generality, mathematical functions have rank 0 0, and many structural functions have rank 1 _. The usual case of v being a mathematical function (left rank 0) means that u . v is equivalent to u@(v"1 _). It is extremely rare to manufacture a use case for dyadic structural functions such as # , {. , but u . # would be equivalent to u@(#"2 _). Though not the strict definition, it is best to think of u . v as "for a 0 (left) rank function v, u . v is shorthand for u@(v"1 _)"