Matrices: Program 1

\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}   % For bold vectors and matrices
\usepackage{amssymb}

\title{Matrix}
\author{}
\date{}

\begin{document}
	
	\maketitle
	
	\section*{Matrix}
	
	\[
	\text{Matrix Multiplication:} \quad \mathbf{A} \mathbf{B} = 
	\begin{pmatrix}
		a_{11} & a_{12} \\
		a_{21} & a_{22}
	\end{pmatrix}
	\begin{pmatrix}
		b_{11} & b_{12} \\
		b_{21} & b_{22}
	\end{pmatrix}
	=
	\begin{pmatrix}
		a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\
		a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22}
	\end{pmatrix}
	\]
	
	\[
	\text{Determinant of a 2x2 Matrix:} \quad \text{det}(\mathbf{A}) = 
	\begin{vmatrix}
		a_{11} & a_{12} \\
		a_{21} & a_{22}
	\end{vmatrix}
	= a_{11}a_{22} - a_{12}a_{21}
	\]
	
	\[
	\text{Inverse of a 2x2 Matrix:} \quad \mathbf{A}^{-1} = \frac{1}{\text{det}(\mathbf{A})}
	\begin{pmatrix}
		a_{22} & -a_{12} \\
		-a_{21} & a_{11}
	\end{pmatrix}
	\]
	
	\[
	\text{Eigenvalue Equation:} \quad \mathbf{A} \bm{v} = \lambda \bm{v}
	\]
	where $\lambda$ is the eigenvalue and $\bm{v}$ is the corresponding eigenvector.
	
	\[
	\text{Diagonalization:} \quad \mathbf{A} = \mathbf{P} \mathbf{D} \mathbf{P}^{-1}
	\]
	where $\mathbf{D}$ is a diagonal matrix, and $\mathbf{P}$ is the matrix of eigenvectors.
	
	\[
	\text{Pauli Matrices:} \quad \sigma_1 = 
	\begin{pmatrix}
		0 & 1 \\
		1 & 0
	\end{pmatrix}, \quad
	\sigma_2 = 
	\begin{pmatrix}
		0 & -i \\
		i & 0
	\end{pmatrix}, \quad
	\sigma_3 = 
	\begin{pmatrix}
		1 & 0 \\
		0 & -1
	\end{pmatrix}
	\]
	
	\[
	\text{Heisenberg Uncertainty Principle (Matrix Form):} \quad \mathbf{X} \mathbf{P} - \mathbf{P} \mathbf{X} = i \hbar \mathbf{I}
	\]
	where $\mathbf{X}$ and $\mathbf{P}$ are position and momentum operators, and $\mathbf{I}$ is the identity matrix.
	
	\[
	\text{Rotation Matrix in 3D:} \quad \mathbf{R}(\theta) = 
	\begin{pmatrix}
		\cos \theta & -\sin \theta & 0 \\
		\sin \theta & \cos \theta & 0 \\
		0 & 0 & 1
	\end{pmatrix}
	\]
	
	\[
	\text{Lorentz Transformation Matrix (Special Relativity):} \quad \mathbf{\Lambda} =
	\begin{pmatrix}
		\gamma & -\gamma \beta & 0 & 0 \\
		-\gamma \beta & \gamma & 0 & 0 \\
		0 & 0 & 1 & 0 \\
		0 & 0 & 0 & 1
	\end{pmatrix}
	\]
	
\end{document}

Mathematical Representations

Matrices: Program 1

Output 1