Changing dimensionality.

For transformations with an equal number of input and output variables ($m=n$), the Jacobian matrices $\textbf{J}_F$ and $\textbf{J}_I$ associated with the forward and inverse mappings (if specified) will both be square. If the transformation functions are correctly formulated, then these two matrices will be mutually inverse and will satisfy:

$$\textbf{J}_F \textbf{J}_I = \textbf{J}_I \textbf{J}_F = \textbf{I}$$ (8)

where $\textbf{I}$ is an identity matrix. Their determinants will also be related by:

$$\det{\textbf{J}_I} = \frac{1}{\det{\textbf{J}_F}}$$ (9)

As a consequence of this (and the definitions of the basic classification properties given below) any property which applies to one of a transformation's two mappings will necessarily apply to the complementary mapping also.

If the transformation affects a change of dimensionality, however, so that $m \ne n$, then it is possible that certain properties may only apply to one of its two mappings. It is still acceptable to associate such properties with the transformation, however, because the TRANSFORM software will take account of the number of input/output variables, and will omit properties which it knows cannot apply when information about a particular mapping is requested. In general, therefore, a classification property may be declared for a transformation if either of its mappings has that property.