Matrix Transformations, Centering, Scaling, and Inverse Methods in Linear Algebra

Matrix Transformations, Centering, Scaling, and Inverse Matrices

By Noorullah Ibrahim

Definition

This lecture focuses on matrix transformations in 2D and higher dimensions, including:

Centering data using matrices
Scaling geometric objects
Combining transformations
Finding matrix inverses
Solving linear systems using inverse matrices

A matrix transformation is a rule that changes vectors or points using matrix multiplication.

Key Points

1. Matrix Transformations

Matrices can perform geometric operations such as:

Rotation
Scaling
Reflection
Translation (indirectly via centering)

Example:

\begin{bmatrix} \cos 45^\circ & \sin 45^\circ \\ -\sin 45^\circ & \cos 45^\circ \end{bmatrix}

rotates a vector.

2. Combining Transformations

Multiple transformations can be combined using matrix multiplication.

Example:

Rotation + Scaling:

\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \cdot \text{Rotation Matrix}

Important rule:

Order matters in matrix multiplication

3. Centering Matrix

Centering shifts data so its mean becomes zero.

Step 1: Compute mean (centroid)

\bar{x} = \frac{1}{n} \sum x_i

Step 2: Subtract mean from each value

X_{centered} = X - \bar{X}

This shifts data to origin (0,0).

4. Scaling Transformation

Scaling changes size of an object.

Multiply by scaling matrix
Example: double size → multiply by 2

2I \cdot X

Steps:

Center object
Multiply by scaling factor

5. 2×2 Matrix Transformations

Used for:

Rotation
Scaling
Reflection

General form:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

6. Matrix Inverse

A matrix inverse behaves like division:

A^{-1}A = I

Where:

( $I$ ) = identity matrix

7. 2×2 Inverse Formula

For:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Inverse:

A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Condition:

( $ad - bc \neq 0$ )

8. Solving Linear Systems Using Inverse

Given:

AX = B

Solution:

X = A^{-1}B

Works only if:

( $A$ ) is invertible

Example / Code

Inverse Matrix Example (2×2)

Given:

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

Inverse:

A^{-1} = \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix}

Verification:

A^{-1}A = I

System of Equations

2x + y = 5

x + 3y = 9

Matrix form:

AX = B

Solution:

X = A^{-1}B

Output (if any)

Identity matrix obtained after inverse multiplication
Unique solution for system of equations if determinant ≠ 0

Common Mistakes

1. Wrong order in multiplication

Matrix multiplication is not commutative:

AB \neq BA

2. Forgetting determinant condition

Inverse exists only if:

ad - bc \neq 0

3. Mixing centering and scaling order

Order changes final result.

4. Incorrect inverse formula usage

Common error: forgetting division by determinant.

Short Exam Notes (very concise revision points)

Centering → subtract mean from data
Scaling → multiply by scalar matrix
Transformation → matrix multiplication
Inverse: ( $A^{-1}A = I$ )
2×2 inverse formula: $\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
Solve system: ( $X = A^{-1}B$ )
Determinant ≠ 0 → inverse exists
Matrix order matters in multiplication