A direction cosine matrix (DCM) is a transformation matrix that transforms one coordinate reference frame to another. If we extend the concept of how the three dimensional direction cosines locate a vector, then the DCM locates three unit vectors that describe a coordinate reference frame. Using the notation in equation 1, we need to find the matrix elements that correspond to the correct transformation matrix.
The first unit vector of the second coordinate frame can be located in the first frame by normal vector notation. See figure 1 for relationship.
Similarily, the other two unit vectors can be described by
It is easy to see how equation 1 works as a transformation matrix through simple matrix multiplication.
Once this transformation matrix is found, it can be used to transform vectors from the second frame to the first frame and vice versa. Equation 2 transforms the x frame to the y frame and can be denoted as . In order to get , which transforms the y frame to the x frame, we use a property of transformation matrices of orthonormal reference frames (a frame that is described by unit vectors and are perpindicular to each other). See the entry on a transformation matrix for more info on its properties. We use the properties that
so using these properties and rearranging equation 2 yields
giving the transformation of the y frame to the x frame
So to extend this concept to transform vectors from one frame to another a closer examination of a vector being represented in both frames is needed. If we denote the second frame as the prime () frame, then a vector expressed in each of these is given by
Since both equations describe the same vector, let us set them equal to each other so
This notation is clumsy so we want to represent it in matrix notation. This is simple enough if you have an understanding of multiplying a column vector by a row vector. This allows us to describe equations 3 and 4 by
Setting them equal and substituting equation 2 in for the second coordinate frame yields
Then by inspection (or go through the matrix manipulation to cancel the x frame)
Representing the transformation matrix as as the transformation from the first frame to the second frame and transposing the previous equation gives
Performing the transposition and using a transposition property for two matrices A and B such that
leads to the relationship
Finally giving us the ability to transform a vector from the second (prime) frame to the first frame.
Much much more can be found in the general entry about the Transformation matrix.