calc/help/mat

Using matrices

    Matrices can have from 1 to 4 dimensions, and are indexed by a
    normal-sized integer.  The lower and upper bounds of a matrix can
    be specified at runtime.  The elements of a matrix are defaulted
    to zeroes, but can be assigned to be of any type.  Thus matrices
    can hold complex numbers, strings, objects, etc.  Matrices are
    stored in memory as an array so that random access to the elements
    is easy.

    Matrices are normally indexed using square brackets.  If the matrix
    is multi-dimensional, then an element can be indexed either by
    using multiple pairs of square brackets (as in C), or else by
    separating the indexes by commas.  Thus the following two statements
    reference the same matrix element:

	    x = name[3][5];
	    x = name[3,5];

    The double-square bracket operator can be used on any matrix to
    make references to the elements easy and efficient.  This operator
    bypasses the normal indexing mechanism, and treats the array as if
    it was one-dimensional and with a lower bound of zero.  In this
    indexing mode, elements correspond to the normal indexing mode where
    the rightmost index increases most frequently.  For example, when
    using double-square bracket indexing on a two-dimensional matrix,
    increasing indexes will reference the matrix elements left to right,
    row by row.  Thus in the following example, 'x' and 'y' are copied
    from the same matrix element:

	    mat m[1:2, 1:3];
	    x = m[2,1];
	    y = m[[3]];

    There are functions which return information about a matrix.
    The 'size' functions returns the total number of elements.
    The 'matdim', 'matmin', and 'matmax' functions return the number
    of dimensions of a matrix, and the lower and upper index bounds
    for a dimension of a matrix.  For square matrices, the 'det'
    function calculates the determinant of the matrix.

    Some functions return matrices as their results.  These	functions
    do not affect the original matrix argument, but instead return
    new matrices.  For example, the 'mattrans' function returns the
    transpose of a matrix, and 'inverse' returns the inverse of a
    matrix.  So to invert a matrix called 'x', you could use:

	    x = inverse(x);

    The 'matfill' function fills all elements of a matrix with the
    specified value, and optionally fills the diagonal elements of a
    square matrix with a different value.  For example:

	    matfill(x,1);

    will fill any matrix with ones, and:

	    matfill(x, 0, 1);

    will create an identity matrix out of any square matrix.  Note that
    unlike most matrix functions, this function does not return a matrix
    value, but manipulates the matrix argument itself.

    Matrices can be multiplied by numbers, which multiplies each element
    by the number.  Matrices can also be negated, conjugated, shifted,
    rounded, truncated, fractioned, and modulo'ed.  Each of these
    operations is applied to each element.

    Matrices can be added or multiplied together if the operation is
    legal.  Note that even if the dimensions of matrices are compatible,
    operations can still fail because of mismatched lower bounds.  The
    lower bounds of two matrices must either match, or else one of them
    must have a lower bound of zero.  Thus the following code:

	    mat x[3:3];
	    mat y[4:4];
	    z = x + y;

    fails because the calculator does not have a way of knowing what
    the bounds should be on the resulting matrix.  If the bounds match,
    then the resulting matrix has the same bounds.  If exactly one of
    the lower bounds is zero, then the resulting matrix will have the
    nonzero lower bounds.  Thus means that the bounds of a matrix are
    preserved when operated on by matrices with lower bounds of zero.
    For example:

	    mat x[3:7];
	    mat y[5];
	    z = x + y;

    will succeed and assign the variable 'z' a matrix whose
    bounds are 3-7.

    Vectors are matrices of only a single dimension.  The 'dp' and 'cp'
    functions calculate the dot product and cross product of a vector
    (cross product is only defined for vectors of size 3).

    Matrices can be searched for particular values by using the 'search'
    and 'rsearch' functions.  They return the element number of the
    found value (zero based), or null if the value does not exist in the
    matrix.  Using the element number in double-bracket indexing will
    then refer to the found element.