Vectors and matrices can be added, subtracted, multiplied, and divided; see Basic Arithmetic.

The `|` (`calc-concat`

) [`vconcat`

] command “concatenates” two vectors
into one. For example, after `[ 1 , 2 ] [ 3 , 4 ] |`, the stack
will contain the single vector ‘`[1, 2, 3, 4]`’. If the arguments
are matrices, the rows of the first matrix are concatenated with the
rows of the second. (In other words, two matrices are just two vectors
of row-vectors as far as `|` is concerned.)

If either argument to `|` is a scalar (a non-vector), it is treated
like a one-element vector for purposes of concatenation: `1 [ 2 , 3 ] |`
produces the vector ‘`[1, 2, 3]`’. Likewise, if one argument is a
matrix and the other is a plain vector, the vector is treated as a
one-row matrix.

The `H |` (`calc-append`

) [`append`

] command concatenates
two vectors without any special cases. Both inputs must be vectors.
Whether or not they are matrices is not taken into account. If either
argument is a scalar, the `append`

function is left in symbolic form.
See also `cons`

and `rcons`

below.

The `I |` and `H I |` commands are similar, but they use their
two stack arguments in the opposite order. Thus `I |` is equivalent
to ` TAB |`, but possibly more convenient and also a bit faster.

The `v d` (`calc-diag`

) [`diag`

] function builds a diagonal
square matrix. The optional numeric prefix gives the number of rows
and columns in the matrix. If the value at the top of the stack is a
vector, the elements of the vector are used as the diagonal elements; the
prefix, if specified, must match the size of the vector. If the value on
the stack is a scalar, it is used for each element on the diagonal, and
the prefix argument is required.

To build a constant square matrix, e.g., a
3x3
matrix filled with ones, use `0 M-3 v d 1 +`, i.e., build a zero
matrix first and then add a constant value to that matrix. (Another
alternative would be to use `v b` and `v a`; see below.)

The `v i` (`calc-ident`

) [`idn`

] function builds an identity
matrix of the specified size. It is a convenient form of `v d`
where the diagonal element is always one. If no prefix argument is given,
this command prompts for one.

In algebraic notation, ‘`idn(a,n)`’ acts much like ‘`diag(a,n)`’,
except that ‘`a`’ is required to be a scalar (non-vector) quantity.
If ‘`n`’ is omitted, ‘`idn(a)`’ represents ‘`a`’ times an
identity matrix of unknown size. Calc can operate algebraically on
such generic identity matrices, and if one is combined with a matrix
whose size is known, it is converted automatically to an identity
matrix of a suitable matching size. The `v i` command with an
argument of zero creates a generic identity matrix, ‘`idn(1)`’.
Note that in dimensioned Matrix mode (see Matrix and Scalar Modes), generic
identity matrices are immediately expanded to the current default
dimensions.

The `v x` (`calc-index`

) [`index`

] function builds a vector
of consecutive integers from 1 to `n`, where `n` is the numeric
prefix argument. If you do not provide a prefix argument, you will be
prompted to enter a suitable number. If `n` is negative, the result
is a vector of negative integers from `n` to *-1*.

With a prefix argument of just `C-u`, the `v x` command takes
three values from the stack: `n`, `start`, and `incr` (with
`incr` at top-of-stack). Counting starts at `start` and increases
by `incr` for successive vector elements. If `start` or `n`
is in floating-point format, the resulting vector elements will also be
floats. Note that `start` and `incr` may in fact be any kind
of numbers or formulas.

When `start` and `incr` are specified, a negative `n` has a
different interpretation: It causes a geometric instead of arithmetic
sequence to be generated. For example, ‘`index(-3, a, b)`’ produces
‘`[a, a b, a b^2]`’. If you omit `incr` in the algebraic form,
‘`index( n, start)`’, the default value for

The `v b` (`calc-build-vector`

) [`cvec`

] function builds a
vector of `n` copies of the value on the top of the stack, where `n`
is the numeric prefix argument. In algebraic formulas, ‘`cvec(x,n,m)`’
can also be used to build an `n`-by-`m` matrix of copies of `x`.
(Interactively, just use `v b` twice: once to build a row, then again
to build a matrix of copies of that row.)

The `v h` (`calc-head`

) [`head`

] function returns the first
element of a vector. The `I v h` (`calc-tail`

) [`tail`

]
function returns the vector with its first element removed. In both
cases, the argument must be a non-empty vector.

The `v k` (`calc-cons`

) [`cons`

] function takes a value `h`
and a vector `t` from the stack, and produces the vector whose head is
`h` and whose tail is `t`. This is similar to `|`, except
if `h` is itself a vector, `|` will concatenate the two vectors
whereas `cons`

will insert `h` at the front of the vector `t`.

Each of these three functions also accepts the Hyperbolic flag [`rhead`

,
`rtail`

, `rcons`

] in which case `t` instead represents
the *last* single element of the vector, with `h`
representing the remainder of the vector. Thus the vector
‘`[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)`’.
Also, ‘`head([a, b, c, d]) = a`’, ‘`tail([a, b, c, d]) = [b, c, d]`’,
‘`rhead([a, b, c, d]) = [a, b, c]`’, and ‘`rtail([a, b, c, d]) = d`’.