Chapter 2: Matrices and Matrix Algebra
Linear Algebra Fundamentals
2.1 Introduction to Matrices
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices provide a powerful way to organize data and solve systems of linear equations.
Definition and Notation
An \(m \times n\) matrix (read as “\(m\) by \(n\)”) has \(m\) rows and \(n\) columns. We denote the element in the \(i\)-th row and \(j\)-th column as \(a_{ij}\).
\[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \]
2.2 Matrix Operations
2.2.1 Matrix Addition and Scalar Multiplication
Addition is performed element-wise and is only defined for matrices of the same size.
Addition: If \(A = [a_{ij}]\) and \(B = [b_{ij}]\), then \(A + B = [a_{ij} + b_{ij}]\). Scalar Multiplication: If \(k\) is a scalar, then \(kA = [k \cdot a_{ij}]\).
2.2.2 Matrix Multiplication
The product \(AB\) is defined only if the number of columns in \(A\) equals the number of rows in \(B\). If \(A\) is \(m \times n\) and \(B\) is \(n \times p\), the resulting matrix \(C = AB\) is \(m \times p\).
The element \(c_{ij}\) is calculated as: \[c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}\]
Table 1: Compatibility of Matrix Operations
| Operation | Requirement | Resulting Dimension |
|---|---|---|
| Addition (\(A+B\)) | \(A\) and \(B\) must be same size (\(m \times n\)) | \(m \times n\) |
| Multiplication (\(AB\)) | Col(\(A\)) = Row(\(B\)) | Row(\(A\)) \(\times\) Col(\(B\)) |
| Transpose (\(A^T\)) | None | \(n \times m\) |
2.3 Special Types of Matrices
- Identity Matrix (\(I\)): A square matrix with 1s on the diagonal and 0s elsewhere. \(AI = IA = A\).
- Zero Matrix (\(0\)): All entries are zero.
- Symmetric Matrix: A matrix where \(A = A^T\).
- Diagonal Matrix: Only the main diagonal contains non-zero entries.
2.4 Real-Life Example: Digital Image Processing
In computer science, a grayscale digital image is essentially a matrix where each element \(a_{ij}\) represents a pixel’s brightness (0 to 255).
Image Transformation (Shear)
Matrix multiplication can be used to rotate, scale, or shear images. Below, we use a shear matrix \(S\) to transform a set of points:
\[ S = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \]
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once per session.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
Figure 1: Geometric Transformation of a Square using Matrix Multiplication
2.5 Real-Life Example: Cryptography
In the Hill Cipher, matrices are used to encrypt text. Letters are converted to numbers (A=0, B=1, etc.), grouped into vectors, and multiplied by an invertible encoding matrix \(E\).
If the message vector is \(P\) (Plaintext), the encrypted vector \(C\) (Ciphertext) is: \[C = E \cdot P \pmod{26}\]
2.6 Exercises
1. Basic Computation Given \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 5 \\ -1 & 2 \end{bmatrix}\), compute: a) \(A + B\) b) \(3A\) c) \(AB\)
2. Transpose Properties Show that for the matrices in Exercise 1, \((AB)^T = B^T A^T\).
3. Application: Inventory Tracking A store has two locations. The inventory for three products is represented by matrices \(L_1\) and \(L_2\): \(L_1 = \begin{bmatrix} 10 & 20 & 30 \end{bmatrix}\) (Items in Location 1) \(L_2 = \begin{bmatrix} 15 & 25 & 10 \end{bmatrix}\) (Items in Location 2) Calculate the total inventory matrix \(T = L_1 + L_2\).
4. Challenge Construct a \(2 \times 2\) matrix \(A\) (other than the Identity or Zero matrix) such that \(A^2 = A\). (This is called an idempotent matrix).
End of Chapter 2 ```