Question

in the theory related to the reproduction of color photography, the equation

is found. the X,Y and Z represent the red,green, and blue densities of the reproductions, respectively, and the x,y,z represent the red,green and blue densities, respectively, of the subject. give the equations relating X,Y, and Z and x,y, and z

Answer 1

This is simple matrix multiplication
$\left[\begin{array}{c}X\\Y\\Z\end{array}\right] = \left[\begin{array}{ccc}1.0&0.1&0.0\\0.5&1.0&0.1\\0.3&0.4&1.0\end{array}\right] \left[\begin{array}{c}x\\y\\z\end{array}\right]$

We have a 3 x 3 matrix (3 rows and 3 columns) and one 3 x 1 matrix (3 rows and 1 column) so the multiplication result will be stored in a 3 x 1 matrix.

Rule : product of MxN, NxP matrix is MxP

To multiply : pick the entire row of first matrix and entire column of second matrix and then multiply values in one-to-one mapping fashion.

$\left[\begin{array}{ccc}1.0&0.1&0.0\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right]$

1. multiply 1.0 to x
2. multiply 0.1 to y
3. multiply 0.0 to z
Finally add all the results = 1.0x + 0.1y + 0.0z = 1.0x + 0.1y

similarly multiply 2nd row, 3rd row

$\left[\begin{array}{ccc}X\\Y\\Z\end{array}\right] = \left[\begin{array}{ccc}1.0x + 0.1y\\0.5x + 1.0y + 0.1z\\0.3x + 0.4y + 1.0z\end{array}\right]$