Transformation Matrix

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.

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 can be classified into different types based on the specific transformations they represent. Some common types of Transformation Matrix include:

• Translation Matrix
• Rotation Matrix
• Scaling Matrix
• Combined Matrix
• Reflection Matrix
• Shear Matrix
• Affine Transformation Matrix

Translation Matrix

A translation matrix is used to shift objects in a coordinate system. Let’s consider a point P(2, 3) and apply a translation of (4, -1) units.

Given point P = (2, 3) and translation vector T = (4, -1), the translation matrix is:

[Tex] \begin{pmatrix} 1 & 0 & 4\\ 0 & 1 & -1\\ 0 & 0 & 1\\ \end{pmatrix} [/Tex]

Applying the translation matrix to point P:

[Tex] \begin{pmatrix} 1 & 0 & 4\\ 0 & 1 & -1\\ 0 & 0 & 1\\ \end{pmatrix} [/Tex] [Tex] \begin{pmatrix} 2\\ 3\\ 1\\ \end{pmatrix} [/Tex] = [Tex] \begin{pmatrix} 6\\ 2\\ 1\\ \end{pmatrix} [/Tex]

Therefore, after the translation, point P(2, 3) is moved to P'(6, 2).

Rotation Matrix

A rotation matrix is used to rotate objects in a coordinate system. Let’s rotate a point Q(1, 1) by 90 degrees counterclockwise.

Given point Q = (1, 1) and rotation angle θ = 90 degrees, the rotation matrix is:

R = [Tex] \begin{pmatrix} 0 & -1\\ 1 & 0\\ \end{pmatrix} [/Tex]

Applying the rotation matrix to point Q:

[Tex]\begin{pmatrix} 0 & -1\\ 1 & 0\\ \end{pmatrix} [/Tex] [Tex]\begin{pmatrix} 1\\ 1\\ \end{pmatrix} [/Tex] = [Tex]\begin{pmatrix} -1 \\ 1\\ \end{pmatrix} [/Tex]

After the rotation, point Q(1, 1) is rotated to Q'(-1, 1).

Scaling Matrix

A scaling matrix is used to resize objects in a coordinate system. Let’s scale a rectangle with vertices A(1, 1), B(1, 3), C(3, 3), D(3, 1) by a factor of 2 in the x-direction and 3 in the y-direction.

Given rectangle ABCD and scaling factors sx = 2, sy = 3, the scaling matrix is:

S = [Tex]\begin{pmatrix} 2 & 0\\ 0 & 3\\ \end{pmatrix} [/Tex]

Applying the scaling matrix to the vertices of the rectangle:

A'(2, 3), B'(2, 9), C'(6, 9), D'(6, 3)

Combined Matrix

A combined matrix applies multiple transformations in sequence. Let’s translate a point P(1, 2) by (3, 4) and then rotate it 45 degrees counterclockwise.

Given point P = (1, 2), translation vector T = (3, 4), rotation angle θ = 45 degrees, the combined matrix is:

C = R⋅T

Applying the combined matrix to point P:

First, translate by T:

T = [Tex]\begin{pmatrix} 1 & 0 & 3\\ 0 & 1 & 4\\ 0 & 0 & 1\\ \end{pmatrix} [/Tex]

P′ = T⋅P = [Tex]\begin{pmatrix} 4\\ 6\\ 1\\ \end{pmatrix} [/Tex]

Then, rotate by R:

R = [Tex]\begin{pmatrix} cos(45) & −sin(45)\\ sin(45) & cos(45)\\ \end{pmatrix} [/Tex]

P′′ = R⋅P′ = [Tex]\begin{pmatrix} -1\\ 5\\ \end{pmatrix}[/Tex]

Reflection Matrix

A reflection matrix is used to mirror objects across a line or plane. Let’s reflect a point Q(2, 3) across the x-axis.

Given point Q = (2, 3), the reflection matrix about the x-axis is:

R_x = [Tex]\begin{pmatrix} 1 & 0\\ 0 & -1\\ \end{pmatrix} [/Tex]

Applying the reflection matrix to point Q:

R_x⋅Q = [Tex]\begin{pmatrix} 2\\ -3\\ \end{pmatrix} [/Tex]

After reflection, point Q(2, 3) is mirrored to Q'(2, -3).

Shear Matrix

A shear matrix is used to skew objects in a coordinate system. Let’s shear a rectangle with vertices A(1, 1), B(1, 3), C(3, 3), D(3, 1) in the x-direction by a factor of 2.

Given rectangle ABCD and shear factor kx = 2, the shear matrix is:

H_x = [Tex]\begin{pmatrix} 1 & 2\\ 0 & 1\\ \end{pmatrix}[/Tex]

Applying the shear matrix to the vertices of the rectangle:

A'(3, 1), B'(3, 3), C'(7, 3), D'(7, 1)

Affine Transformation Matrix

An affine transformation matrix combines linear transformations with translations. Let’s apply an affine transformation to a point P(1, 1) by scaling it by a factor of 2 in the x-direction, rotating it 30 degrees counterclockwise, and then translating it by (2, 3).

Given point P = (1, 1), scaling factor sx = 2, rotation angle θ = 30 degrees, translation vector T = (2, 3), the affine transformation matrix is:

A = R⋅S⋅T

Applying the affine transformation matrix to point P:

First, scale by S:

S = [Tex]\begin{pmatrix} 2 & 0\\ 0 & 1\\ \end{pmatrix} [/Tex]

P′ = S⋅P = [Tex]\begin{pmatrix} 2\\ 1\\ \end{pmatrix} [/Tex]

Then, rotate by R:

R = [Tex]\begin{pmatrix} cos(30) & −sin(30)\\ sin(30) & cos(30)\\ \end{pmatrix} [/Tex]

P′′ = R⋅P′ = [Tex]\begin{pmatrix} 1.732\\ 1.5\\ \end{pmatrix} [/Tex]

Finally, translate by T:

T = [Tex]\begin{pmatrix} 1 & 0 & 2\\ 0 & 1 & 3\\ 0 & 0 & 1\\ \end{pmatrix} [/Tex]

P′′′ =T⋅P′′ = [Tex]\begin{pmatrix} 3.732\\ 4.5\\ \end{pmatrix} [/Tex]

After the affine transformation, point P(1, 1)

Transformation matrices have numerous applications in various fields, including:

• Computer Graphics: Used for rendering 3D scenes, modeling objects, and applying transformations to vertices.

• Image Processing: Applied for image warping, distortion correction, and geometric transformations.

• Robotics: Used in the determination of geometric properties of the end-effectors of robotic manipulators.

• Geometric Modeling: An important part of both CAD/CAM systems in terms of creating and modifying shapes, surfaces, and solids is an element which is known as parametric.

• Mathematics and Physics: Apply in the study of linear transformation, vector space and coordinate systems.

Transformation matrix is a great tool for linear algebra, it is a compact and convenient way to realize and implement various transformations to both vectors and points. It is possible to move figures by merely recognizing different transformation matrices and their properties. It is possible to generate interior scenes intricately and also to solve problems of different fields by transforming objects. The applications of transformation matrices extend beyond the examples mentioned here, making them an essential concept in mathematics and its applications.

Read More:

• Null Matrix
• Row Matrix
• Column Matrix
• Orthogonal Matrix
• Idempotent Matrix
• Nilpotent Matrix

Example 1: Find the new matrix after transformation using the transformation matrix [Tex]\begin{pmatrix} 2 & -3\\ 1 & 2\\ \end{pmatrix} [/Tex]on the vector A = 5i + 4j.

Solution:

Given transformation matrix is T = [Tex]\begin{pmatrix} 2 & -3\\ 1 & 2\\ \end{pmatrix} [/Tex]

Given vector A = 5i + 4j is written as a column matrix as A = [Tex]\begin{pmatrix} 5\\ 4\\ \end{pmatrix} [/Tex]

Let new matrix after transformation be B, and we have the transformation formula as TA = B

B = TA = [Tex]\begin{pmatrix} 2 & -3\\ 1 & 2\\ \end{pmatrix} [/Tex] x [Tex]\begin{pmatrix} 5\\ 4\\ \end{pmatrix} [/Tex]

B = [Tex]\begin{pmatrix} 2 * 5 + (-3) * 4\\ 1 * 5 + 2 * 4\\ \end{pmatrix} [/Tex]

B = [Tex]\begin{pmatrix} -2\\ 13\\ \end{pmatrix} [/Tex]

B = -2i + 13j

Therefore, the new matrix on transformation is -2i + 13j

Example 2: Find the value of the constant ‘a’ in the transformation matrix [Tex]\begin{pmatrix} 1 & a\\ 0 & 1\\ \end{pmatrix}[/Tex] , which has transformed the vector A = 3i + 2j to another vector B = 7i + 2j.

Solution:

Given vectors are A = 3i + 2j and B = 7i + 2j

These vectors written as column matrices are equal to A = [Tex]\begin{pmatrix} 3\\ 2\\ \end{pmatrix} [/Tex], and B = [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix} [/Tex]

This is shear transformation, where only one component of the matrix is changed.

Given transformation matrix is T = [Tex]\begin{pmatrix} 1 & a\\ 0 & 1\\ \end{pmatrix}[/Tex]

Applying the formula of transformation matrix, TA = B, we have the following calculations:

[Tex]\begin{pmatrix} 1 & a\\ 0 & 1\\ \end{pmatrix}[/Tex] x [Tex]\begin{pmatrix} 3\\ 2\\ \end{pmatrix} [/Tex]= [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix} [/Tex]

[Tex]\begin{pmatrix} 1 × 3 + a × 2\\ 0 × 3 + 1 × 2\\ \end{pmatrix} [/Tex]= [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix}[/Tex]

[Tex]\begin{pmatrix} 3 + 2a\\ 2\\ \end{pmatrix}[/Tex] = [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix}[/Tex]

Comparing the elements of the above two matrices, we can calculate the value of a:

3 + 2a = 7

2a = 7 – 3

2a = 4

a = 4/2 = 2

Therefore, the value of a = 2, and the transformation matrix is [Tex]\begin{pmatrix} 1 & 2\\ 0 & 1\\ \end{pmatrix} [/Tex]

What is a Transformation Matrix?

Transformation matrix is a square matrix that is representative of the linear transformation’s vectors in the vector space. The vectors’ shapes, sizes, or orientations can be altered by this transformation.

How to Apply a Transformation Matrix?

To apply a transformation matrix, you multiply it by a vector representing a point or object in space, resulting in a new vector that represents the transformed point or object.

What are Basic Elementary Operations of a Matrix?

Basic elementary operations of a matrix include row operations (like swapping rows, multiplying a row by a scalar, or adding multiples of one row to another) and column operations.

Is Transformation Matrix Left or Right?

A transformation matrix can be either left or right, depending on whether it’s multiplied on the left or right side of the vector representing the point or object to be transformed.

What is the Range of a Transformation Matrix?

Range of a transformation matrix represents all possible output vectors that can be obtained by applying the matrix to input vectors.