# Vocabulary/dotdot

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`u .. v`Even Conjunction

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

`u .: v`Odd Conjunction

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

(`u .. v`) is the same as `(u + u&v) % 2:`

(`u .: v`) is the same as `(u - u&v) % 2:`

u .. vandu .: varedeprecated.

Replace the functions (seldom used) by their equivalent phrases above.

Future releases of J may reassign the words..and.:

### Common uses

1. Make a function out of `u` which is symmetrical about zero on the X-axis, using `-` for `v`

**Even** (`..-`) gives a symmetrical function, **Odd** (`.:-`) gives an antisymmetrical function.

u=: ^ NB. exponential growth: sample unsymmetrical function ] X=: 5 %~ i:5 _1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1 require 'plot' plot X; u X

plot X; u ..- X

plot X; u .:- X

2. Decompose a matrix into symmetric and antisymmetric parts, or Hermitian and antiHermitian parts, using `|:` for `v`

]a =. i. 3 3 0 1 2 3 4 5 6 7 8 ]asymm =: ] .. |: a 0 2 4 2 4 6 4 6 8 ]aantisymm =: ] .: |: a 0 _1 _2 1 0 _1 2 1 0 asymm + aantisymm 0 1 2 3 4 5 6 7 8

For complex matrices let `v` be `+@:|:`, to take the adjoint.

### Details

- . Any mathematical function can be uniquely decomposed as the sum of an even and an odd function.
- . Any matrix can be uniquely decomposed as the sum of a symmetric and an antisymmetric matrix.
- . Any matrix can be uniquely decomposed as the sum of a Hermitian and an antiHermitian matrix.