Unification

Augment each point to be a 3D row vector by adding a third unit coordinate :

All 5 transformations can then be performed using a 3x3 array of 9 coefficients

with m[2][0] = 0; m[2][1] = 0; m[2][2] = 1;
and the remaining 6 coefficients doing all the work.

A transformation can then be abbreviated to :

A property of this unification is that if

then

The elements of M₃₁ can be derived from those of M₃₂ and M₂₁ in a simple fashion:

In mathematics, M is called a matrix , and the combining of two such matrices in this way is called matrix multiplication.

This is very important for computer graphics. For scenes containing many points being viewed by a series of transformations, each point need only be transformed by one combined matrix instead of a succession of matrices.

Inverse Transformations

Each of the 5 basic transformations has an inverse, written M^-1 :

Note that if M₁ = M₂^-1 then M₂ = M₁^-1

The product of a matrix with its inverse is called the Unit or Identity matrix :

Similarity Transformations

In a triple matrix product: if the first and third transformations are the inverse of each other, the the second transformation is said to be undergoing a similarity transformation, eg

Rows versus Columns

If the vector had been written as a column instead of a row:

then the rows and columns need to be interchanged (transposed) in the previous algebra, and matrices are applied to the left of a vector instead of the right :

and matrices are concatenated on the left instead of the right :

A matrix that is the same when its rows and columns are transposed is said to be symmetric.
For example the matrices for scaling and reflection are symmetric.
The transpose of a rotation matrix is its inverse.