**2.1.2. Linear transformation matrix**

The proposal to replace the `CROTA i` keywords of Wells et
al.
([1981])
with a general linear transformation matrix dates
from Hanisch & Wells
([1988]),
although the details of its
implementation have undergone considerable evolution. The main point
of divergence has been whether the matrix should completely replace or
simply augment the

In defining a nomenclature which augments the `CDELT i` we
have been guided by the following considerations:

- Where possible, standards should grow by
generalizing existing usage rather than developing a separate
parallel usage. Augmenting the existing
`CDELT`with a separate transformation matrix that defaults to unity makes old headers equivalent to new ones that omit the keywords that define the transformation matrix. In any case, the "once FITS, always FITS" rule means that FITS readers must continue to interpret*i*`CDELT`, so it makes sense for*i*`CDELT`to retain its original function.*i* - The transformation matrix then replaces the
poorly defined
`CROTA`with a nomenclature that allows for both skew andfully general rotations. We do not consider this replacement and the consequent deprecation of the*i*`CROTA`to be inconsistent with the aim of generalizing existing usage since, to our knowledge, the*i*`CROTA`have had no formal definition other than the "AIPS convention" (Greisen [1983], [1986]). Both Wells et al. ([1981]) and Hanisch et al. ([2001]) state that "users of this option should provide extensive explanatory comments." Paper II describes the translation of the AIPS interpretation of*i*`CROTA`to the new formalism.*i* - A large fraction of WCS representations, perhaps
the great majority, will not require the general linear
transformation. FITS writers may continue to use
`CDELT`, so FITS-writing software need not be rewritten to conform to the new formalism unless it needs the new features.*i* - The physical units of a general image may differ by
many orders of magnitude, from frequencies of 10
^{10}Hz (or more) to angles of 10^{-3}degrees (or less). If the physical units enter into the linear transformation matrix, then the elements of that matrix will have very different magnitudes. These issues pose difficulties both in computing and in understanding, and it may be simpler to defer application of physical units until the multiplication by`CDELT`.*i* - These difficulties are compounded when correcting
for the distortions present in real instruments. Paper IV will show
that some instruments require distortion corrections before, and
others after, the linear transformation matrix. Such corrections
may need to be expressed in terms directly related to pixel
coordinates. If the physical units enter into the linear
transformation matrix, then the distortion corrections which come
after the matrix would have to compensate for the physical units
applied by it, effectively undoing and then redoing a multiplication
by
`CDELT`. Furthermore, commensurability problems may arise when recording the maximum distortion correction for a WCS representation that mixes pre-, and post-corrections between axes.*i* - A widely used formalism that discards
`CDELT`was developed by the Space Telescope Science Institute for the Hubble Space Telescope and was incorporated generally in the IRAF data analysis system. We therefore support this as an alternative method.*i*

In the `PC i_j` formalism, the matrix elements

PC i_j |
(floating value) |

header cards, and *s*_{i} as `CDELT i`. The

In the `CD i_j` formalism Eqs. (1) and (2) are
combined as

(3) |

and the

CD i_j |
(floating-valued) |

keywords encode the product
*s*_{i}*m*_{ij}. The *i* and *j*
indices are used without leading zeroes, e.g. `CD1_1`. The
`CD i_j` matrix must not be singular; it must have
an inverse.

We specifically prohibit mixing of the `PC i_j` and