Chapter 18: lojbau mekso: Mathematical Expressions in Lojban

15. Vectors and matrices

The following cmavo are discussed in this section:

jo'i    JOhI    start vector
te'u    TEhU    end vector
pi'a    VUhU    matrix row combiner
sa'i    VUhU    matrix column combiner

A mathematical vector is a list of numbers, and a mathematical matrix is a table of numbers. Lojban considers matrices to be built up out of vectors, which are in turn built up out of operands.

“jo'i”, the only cmavo of selma'o JOhI, is the vector indicator: it has a syntax reminiscent of a forethought operator, but has very high precedence. The components must be simple operands rather than full expressions (unless parenthesized). A vector can have any number of components; “te'u” is the elidable terminator. An example:

✥15.1    li jo'i paboi reboi te'u su'i jo'i ciboi voboi
    du li jo'i voboi xaboi
the-number array( one, two ) plus array( three, four)
    equals the-number array( four, six)
(1,2) + (3,4) ⇔ (4,6)

Vectors can be combined into matrices using either “pi'a”, the matrix row operator, or “sa'i”, the matrix column operator. The first combines vectors representing rows of the matrix, and the second combines vectors representing columns of the matrix. Both of them allow any number of arguments: additional arguments are tacked on with the null operator “ge'a”.

Therefore, the “magic square” matrix

8 1 6 3 5 7 4 9 2
can be represented either as:

✥15.2    jo'i biboi paboi xa pi'a jo'i ciboi muboi ze
    ge'a jo'i voboi soboi re
the-vector (8 1 6) matrix-row the-vector (3 5 7)
    , the-vector (4 9 2)

or as

✥15.3    jo'i biboi ciboi vo sa'i jo'i paboi muboi so
    ge'a jo'i xaboi zeboi re
the-vector (8 3 4) matrix-column the-vector (1 5 9)
    , the-vector (6 7 2)

The regular mekso operators can be applied to vectors and to matrices, since grammatically both of these are expressions. It is usually necessary to parenthesize matrices when used with operators in order to avoid incorrect groupings. There are no VUhU operators for the matrix operators of inner or outer products, but appropriate operators can be created using a suitable symbolic lerfu word or string prefixed by “ma'o”.

Matrices of more than two dimensions can be built up using either “pi'a” or “sa'i” with an appropriate subscript numbering the dimension. When subscripted, there is no difference between “pi'a” and “sa'i”.