What is a Transformation Matrix?

A transformation matrix is a square matrix, which represents a linear transformation in vector space. It transforms (from one) coordinated system to a (different) system by keeping the nature of that space identical. Therefore, it functions by keeping the linearity attribute of the space. Alternatively, it is a linear transformation in which the matrix carries the coefficients making it easier to compute and change the geometrical objects.

For illustration, look at a 2D coordinate system with coordinate vectors i and j. A transformation matrix T can be utilized to take a vector v = (x, y) and transform it to a vector w = (x’, y’) which forms a new coordinate system. The transformation matrix T consists of the coefficients that determine the directions that in the basis vector i and j are transformed to and these coefficients are used to find the vector w.

Properties of Transformation Matrix

Various properties of the Transformation Matrix are:

• Transformation matrices are square matrices, which have the number of rows and columns equal to the extent of dimensions of the vector space.

• The product of a single transformation matrix can represent the composite of the corresponding linear transformations, accordingly.

• Identity matrix is a special transformation matrix that represents the identity transformation, where every vector is mapped to itself.

• Invertible transformation matrices have a unique inverse matrix that undoes the transformation.

• Transformation matrices can be combined through matrix multiplication to create more complex transformations.

Transformation matrices are the core notions in linear algebra and these can help make advancements in many areas including computer graphics, image processing, and so on. Zero vectors and the corresponding unit vectors provide a compact and generalized manner of applying transformations to vectors or points in a coordinate system.

In this article, we will explore detail about transformation matrices, their basic principles, types various applications and others in detail.