Color Transformations

 In matrix notation,

Using (1.8.20)

 Therefore,

M	(2.9)

Thus, the tristimulus values can be transformed between the two primaries by knowing matrix M. In particular, the color matching functions corresponding to the two sets of primaries are related by,

	(2.10)

The chromaticity coordinates of a color C can also be transformed between primary systems. For example, if we define sums of tristimulus values of C by

Then previous equation (2.9) can be written in terms of chromaticity coordinates as

	(2.11)

Rewriting and employing the matrix element summation operator  we get

	=

Thus,

	(2.12)

which gives the rgb chromaticity coordinates in terms of the xyz chromaticity coordinates

and the transformation matrix M. A similar relation can be derived for xyz in terms of rgb.

In some cases, only the matrix m of chromaticity coordinates relating the two primary systems is given. To handle this case we now derive a relation between M and m. Let the sum of the tristimulus values be defined as below:

Substituting the relationships between chromaticity coordinates and tristimulus values

(e.g., ) into Eq.(2.3), we get

	(2.13)

Let S = Diag  be a  diagonal matrix. Then

	(2.14)

Thus for an arbitrary color C, substituting forM in  Eq. (2.9) 

For unit intensity equal energy white, all the tristimulus values are unity, and therefore we must have

Sm
m	(2.16)
m -1

Thus, from Eq. (2.14) and the definition of S

M = Diag

and we have derived the relationship between matrices m and M,which can then be used to convert arbitary colors between the two primary systems.