Transformation Matrix

Transformation Matrix is a matrix that transforms one vector into another vector by the process of matrix multiplication. The transformation matrix alters the cartesian system and maps the coordinates of the vector to the new coordinates. The transformation matrix T of order m x n on multiplication with a vector A of n components represented as a column matrix transforms it into another matrix representing a new vector A'.

For a two-dimensional vector space, the transformation matrix is of order 2 x 2, and for an n-dimensional space, the transformation matrix is of order n x n. Let us learn more about the transformation matrix, types of transformation matrices, and application of the transformation matrix.

Transformation matrix is a matrix that transforms one vector into another vector. The positional vector of a point is changed to another positional vector of a new point, with the help of a transformation matrix. The position vector of a point A = xi + yj, on multiplying with a matrix T = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is transformed to another vector B = x'i + y'j. Here the vector A = xi + yj is represented as a column matrix A = \(\begin{bmatrix}x\\y\end{bmatrix}\), and the matrix B = x'i + y'j is another column matrix B = \(\begin{bmatrix}x'\\y'\end{bmatrix}\).

The operation of the transformation of vectors using a transformation matrix is as follows.

TA = B

\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)

The transformation matrix can be taken as the transformation of space. Here a 2 x 2 transformation matrix is used for two-dimensional space, and a 3 x 3 transformation matrix is used for three-dimensional space. The vector in a two-dimensional space has two components which is represented as a column vector with two elements, and following the matrix multiplication condition it is transformed using a transformation matrix of order 2 x 2.

The transformation matrix transforms a vector into another vector, which can be understood geometrically in a two-dimensional or a three-dimensional space. The frequently used transformations are stretching, squeezing, rotation, reflection, and orthogonal projection. Let us learn about some of these transformations in detail.

Stretching

The linear transformation enlarges the distance in the xy plane by a constant value. Here the distance is enlarged or compressed in a particular direction with reference to only one of the axis and the other axis is kept constant. A stretch along the x-axis by keeping the y-axis the same is x' = kx, and y' = y. Here k is a constant numeric value and for k > 1 it is a stretch, k < 1 it is a compression, and for k = 1 it is the same point. The transformation matrix for a stretch along the x-axis is as follows.

T_x = \(\begin{bmatrix}k&0\\0&1\end{bmatrix}\)

Also, we can stretch along the y-axis to obtain x' = x, and y' = ky. The transformation matrix for a stretch along the y-axis is as follows.

T_y = \(\begin{bmatrix}k&0\\0&1\end{bmatrix}\)

Rotation

The transformation matrix helps to rotate the vector in an anticlockwise direction at an angle θ. The transformation matrix \(\begin{bmatrix}Cosθ&Sinθ\\-Sinθ = &Cosθ\end{bmatrix}\) transforms the vector xi + yj to x'i + y'j, which is represented as follows.

\(\begin{bmatrix}Cosθ&Sinθ\\-Sinθ&Cosθ\end{bmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)

Shearing

The shearing changes the vector in such a way that the square boxes in the coordinate axes are deformed into parallelograms. The transformation matrix for shearing along the x-axis is \(\begin{bmatrix}1&k\\0&1\end{bmatrix}\) and it transforms the vector A = xi + yj to A' = x'i + y'j such that x' = x + ky, y' = y. This transformation can be understood from the below multiplication of matrices.

\(\begin{bmatrix}1&k\\0&1\end{bmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)

Also, the transformation matrix for shearing along the y-axis is \(\begin{bmatrix}1&0\\k&1\end{bmatrix}\), and it changes changes the vector such that x' = x, and y' = y + kx. The matrix multiplication for this transformation is as follows.

\(\begin{bmatrix}1&0\\k&1\end{bmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)

The transformation matrix has numerous applications in vectors, linear algebra, matrix operations. The following are some of the important applications of the transformation matrix.

• Vectors represented in a two or three-dimensional frame are transformed to another vector.
• Linear Combinations of two or more vectors through multiplication are possible through a transformation matrix.
• The linear transformations of matrices can be used to change the matrices into another form.
• Matrix multiplication is the transformation of one matrix into another matrix.
• Determinants can be solved using the concepts of the transformation matrix.
• Inverse Space also use matrix transformations.
• Dot Product and Cross Product of Vectors
• Change of Basis of vectors is possible through transformations
• Eigen Vectors and Eigen Values involve matrix and matrix transformation.
• Abstract Vector Spaces also use the concepts of the transformation matrix.

Related Topics

The following topics help for a better understanding of the transformation matrix.

• Matrix Operations
• Singular Matrix
• Unitary Matrix
• Hermitian Matrix
• Skew Symmetric Matrix

How Is Transformation Matrix?

Transformation Matrix is used to transform one vector into another vector by the process of matrix multiplication. The position vector of a point is represented as a column matrix, and the number of elements in this column matrix is equal to the components of the vector. The multiplication of a transformation matrix with the column matrix of the vector gives a new matrix of the transformed vector.

What Is the Transformation Matrix Formula?

The formula for transformation matrix is TA = A'. The transformation matrix T = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) on multiplication with a position vector A = xi + yj represented as a column matrix \(\begin{bmatrix}x\\y\end{bmatrix}\), transforms it into another matrix \(\begin{bmatrix}x\\y\end{bmatrix}\), representing a new matrix with position vector A' = x'i + y'j. The transformation matrix formula can be represented in the following matrix form.

\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)

How Does Transformation Matrix Work?

The transformation matrix works with the formula of matrix multiplication. The multiplication of the transformation matrix T = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\), with a column matrix A = \(\begin{bmatrix}x\\y\end{bmatrix}\), results in a new matrri A' = \(\begin{bmatrix}x'\\y'\end{bmatrix}\). The condition for matrix multiplication should match before performing the multiplication of transformation matrix. The number of columns in the transformation matrix T should be equal to the number of elements or rows in the column matrix A.

What Are the Types of Transformation Matrix?

The type transformation matrix depends on the transformation which they can perform on the vector in a two-dimensional or three-dimensional space. The frequently performed transformations using a transformation matrix are stretching, squeezing, rotation, reflection, and orthogonal projection.

What Are the Uses of Transformation Matrix?

The following are some of the important uses of the transformation matrix.

• Vectors represented in a two or three-dimensional frame are transformed to another vector.
• Linear Combinations of two or more vectors through multiplication are possible through a transformation matrix.
• The linear transformations of matrices can be used to change the matrices into another form.
• Matrix multiplication is the transformation of one matrix into another matrix.