Matrix and Scalar Modes
-----------------------

Calc sometimes makes assumptions during algebraic manipulation that
are awkward or incorrect when vectors and matrices are involved.  Calc
has two modes, "matrix mode" and "scalar mode", which modify its
behavior around vectors in useful ways.

Press `m v' (`calc-matrix-mode') once to enter matrix mode.  In this
mode, all objects are assumed to be matrices unless provably
otherwise.  One major effect is that Calc will no longer consider
multiplication to be commutative.  (Recall that in matrix arithmetic,
`A*B' is not the same as `B*A'.)  This assumption affects rewrite
rules and algebraic simplification.  Another effect of this mode is
that calculations that would normally produce constants like 0 and 1
(e.g., `a - a' and `a / a', respectively) will now produce function
calls that represent "generic" zero or identity matrices: `idn(0)',
`idn(1)'.  The `idn' function `idn(A,N)' returns A times an NxN
identity matrix; if N is omitted, it doesn't know what dimension to
use and so the `idn' call remains in symbolic form.  However, if this
generic identity matrix is later combined with a matrix whose size is
known, it will be converted into a true identity matrix of the
appropriate size.  On the other hand, if it is combined with a scalar
(as in `idn(1) + 2'), Calc will assume it really was a scalar after
all and produce, e.g., 3.

Press `m v' a second time to get scalar mode.  Here, objects are
assumed *not* to be vectors or matrices unless provably so.  For
example, normally adding a variable to a vector, as in `[x, y, z] +
a', will leave the sum in symbolic form because as far as Calc knows,
`a' could represent either a number or another 3-vector.  In scalar
mode, `a' is assumed to be a non-vector, and the addition is evaluated
to `[x+a, y+a, z+a]'.

Press `m v' a third time to return to the normal mode of operation.

If you press `m v' with a numeric prefix argument N, you get a special
"dimensioned matrix mode" in which matrices of unknown size are
assumed to be NxN square matrices.  Then, the function call `idn(1)'
will expand into an actual matrix rather than representing a "generic"
matrix.

Of course these modes are approximations to the true state of
affairs, which is probably that some quantities will be matrices
and others will be scalars.  One solution is to "declare"
certain variables or functions to be scalar-valued.
See Declarations, to see how to make declarations in Calc.

There is nothing stopping you from declaring a variable to be scalar
and then storing a matrix in it; however, if you do, the results you
get from Calc may not be valid.  Suppose you let Calc get the result
`[x+a, y+a, z+a]' shown above, and then stored `[1, 2, 3]'
in `a'.  The result would not be the same as for `[x, y, z]
+ [1, 2, 3]', but that's because you have broken your earlier promise
to Calc that `a' would be scalar.

Another way to mix scalars and matrices is to use selections (*Note
Selecting Subformulas::).  Use matrix mode when operating on your
formula normally; then, to apply scalar mode to a certain part of the
formula without affecting the rest just select that part, change into
scalar mode and press `=' to resimplify the part under this mode, then
change back to matrix mode before deselecting.