Matrix Transformations

Matrix Transformations

Basic Principles

• A Matrix Transformation is a linear transformation that can be associated with a matrix.
• It is a rule that takes an input vector and transforms it into an output vector.
• This is done by multiplying the input vector (also known as pre-multiplication) or output vector (post-multiplication) by the corresponding matrix.

Types of Transformations

• A Translation moves every point a fixed distance in a given direction, and it is not representable by a matrix.
• A Scaling transformation changes the size of an object. This operation can be represented by a diagonal matrix with the scale factor along the main diagonal.
• A Rotation changes the orientation of an object. This transformation is represented by a special class of matrices known as rotation matrices.
• Reflection flips an object across a line. Reflection is represented by so-called reflection matrices.
• Shearing, also known as skewing, distorts an object shape in a fixed direction while keeping the lines of the object parallel. It is also represented by certain matrices.

Properties of Matrix Transformations

• Transformations are characterised by the Linearity Principle which states that the sum of the transformations equals the transformation of the sum, and the transformation of a scalar multiplication equals the scalar multiplication of the transformation.
• Transformations preserve the parallellism of lines, as well as the midpoint of line segments.
• The determinant of the transformation matrix can tell us a lot about the transformation itself. If the determinant is positive, the transformation preserves orientation. If it is negative, the transformation reverses orientation.

Composition of Transformations

• The composition of two matrix transformations is another matrix transformation, and corresponds to the matrix multiplication of the two transformation matrices.
• The order of composition matters: in general, transforming by matrix A then matrix B is not the same as transforming by B then A. This is due to the non-commutative nature of matrix multiplication.
• When multiple transformations are applied in sequence, the final effect is found by multiplying the matrices together in the order in which their transformations were applied.

Inverse Transformation

• The inverse of a transformation, if it exists, ‘undoes’ the effect of the transformation.
• It is represented by the inverse of the transformation matrix.
• If the determinant of a matrix is 0, the matrix is called singular, and it does not have an inverse. Correspondingly, the transformation does not have an inverse.