APSC 174: Linear Algebra for Engineers

Definition

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Formally, an $m \times n$ matrix $A$ has $m$ rows and $n$ columns:

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}

Augmented Matrices as Systems of Linear Equations

Matrix Representation

Types of Matrices

Square Matrix: A matrix with the same number of rows and columns ( $m = n$ )
Identity Matrix: A square matrix with 1's on the main diagonal and 0's elsewhere
Augmented Matrix: A matrix used to represent a system of linear equations
Zero Matrix: A matrix with all elements equal to zero
Diagonal Matrix: A square matrix with non-zero elements only on the main diagonal
Triangular Matrix:
- Upper triangular: non-zero elements only on and above the main diagonal
- Lower triangular: non-zero elements only on and below the main diagonal
Symmetric Matrix: A square matrix equal to its transpose ( $A = A^T$ )

Matrix Operations

Addition and Subtraction

For matrices $A$ and $B$ of the same dimensions:

(A \pm B)_{ij} = A_{ij} \pm B_{ij}

Scalar Multiplication

For a scalar $c$ and matrix $A$ :

(cA)_{ij} = c \cdot A_{ij}

Matrix Multiplication

For an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ :

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj}

Transpose

The transpose of an $m \times n$ matrix $A$ is the $n \times m$ matrix $A^T$ where:

(A^T)_{ij} = A_{ji}

Properties and Applications

Determinant

For a square matrix, the determinant is a scalar value that provides information about the matrix's invertibility and the volume scaling factor of the linear transformation it represents.

Inverse

A square matrix $A$ has an inverse $A^{-1}$ if $AA^{-1} = A^{-1}A = I$ , where $I$ is the identity matrix.

Rank

The rank of a matrix is the dimension of the vector space generated by its columns (or rows).

Eigenvalues and Eigenvectors

For a square matrix $A$ , a non-zero vector $v$ is an eigenvector with eigenvalue $\lambda$ if $Av = \lambda v$ .

Applications

Systems of Linear Equations: Matrices provide a compact way to represent and solve systems using methods like Gaussian elimination and Row-Reduced Echelon Form.
Linear Transformations: Matrices represent linear transformations between vector spaces.
Computer Graphics: Transformation matrices are used for rotation, scaling, and translation in 2D and 3D graphics.
Data Science: Matrices are fundamental in techniques like Principal Component Analysis (PCA) and Linear Regression.
Quantum Mechanics: Matrices represent observables and transformations in quantum systems.

Computational Methods

Various algorithms exist for matrix operations, including:

LU Decomposition
QR Factorization
Singular Value Decomposition (SVD)
Eigenvalue Decomposition