Matrices

Definition

An $m \times n$ matrix is a rectangular array:

$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$

Element $a_{ij}$ is in row $i$, column $j$.

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Basic Operations

Addition

For $m \times n$ matrices $A$ and $B$:

$(A + B)_{ij} = a_{ij} + b_{ij}$

Scalar Multiplication

$(\lambda A)_{ij} = \lambda a_{ij}$

Matrix Multiplication

For $A$ ($m \times p$) and $B$ ($p \times n$):

$(AB)_{ij} = \sum_{k=1}^{p} a_{ik}b_{kj}$

Condition: Number of columns of $A$ = number of rows of $B$.

Warning: Matrix multiplication is not commutative: $AB \neq BA$ in general.

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Transpose

$A^T_{ij} = A_{ji}$

Properties:

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(AB)^T = B^T A^T$ (reversed order)

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Trace

The trace of a square $n \times n$ matrix:

$\text{Tr}(A) = \sum_{i=1}^{n} a_{ii}$

Properties:

• $\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B)$
• $\text{Tr}(\lambda A) = \lambda\,\text{Tr}(A)$
• $\text{Tr}(AB) = \text{Tr}(BA)$ (cyclic property)

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Identity Matrix

$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$

$AI = IA = A$ for compatible matrices.

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Special Types

Type	Definition
Diagonal	$a_{ij} = 0$ for $i \neq j$
Upper triangular	$a_{ij} = 0$ for $i > j$
Lower triangular	$a_{ij} = 0$ for $i < j$
Symmetric	$A^T = A$
Antisymmetric	$A^T = -A$
Orthogonal	$A^T A = I$

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Complex Matrices

Complex Conjugate

$(A^*)_{ij} = \bar{a}_{ij}$

Hermitian Conjugate

$A^\dagger = (A^T)^*$

Hermitian Matrix

$A^\dagger = A$

Hermitian matrices have real eigenvalues (important in quantum mechanics).

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Representing Linear Maps

A matrix $A$ ($m \times n$) represents a linear map $\mathbb{R}^n \to \mathbb{R}^m$:

$\mathbf{y} = A\mathbf{x}$

where $\mathbf{x} \in \mathbb{R}^n$ (column vector) and $\mathbf{y} \in \mathbb{R}^m$.

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Example: Rotation Matrix

2D rotation by angle $\theta$:

$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$

Properties:

• $R(\theta)R(\phi) = R(\theta + \phi)$
• $R(\theta)^{-1} = R(-\theta)$
• $\det(R) = 1$
• Orthogonal: $R^T R = I$

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Why Matrices Matter

• Represent linear systems of equations compactly
• Encode transformations (rotations, scaling, reflections)
• Essential for computer graphics and physics simulations
• Foundation for quantum mechanics (operators)