Linear Algebra With Eigen and C++

This repository contains my tutorials on mastering Matrix operation and numerical optimization with Eigen and C++. Every chapter below is a written tutorial paired with a self-contained, compilable example in src/ — from basic matrix arithmetic up to SVD-based camera calibration, point-cloud registration and sparse SLAM pose-graph solving.

Build and run

Requires a C++17 compiler, CMake >= 3.21, Ninja and Eigen 3 (sudo apt install libeigen3-dev on Debian/Ubuntu).

git clone https://github.com/behnamasadi/EigenDemo.git
cd EigenDemo
cmake --preset ninja-multi
cmake --build build --config Release

The binaries land in build/Release/, one per example, so you can run any topic directly:

./build/Release/singular_value_decomposition
./build/Release/quaternion
./build/Release/slam_pose_graph

The following is the outline of this repository:

Chapter 1 Introduction and Installation

• About Eigen
• Installation
• Adding Eigen to Your Project
• Your First Eigen Program

Chapter 2 Matrix, Array and Vector Class

• Matrix Class
• Vector Class
• Array Class
• Initialization
• Accessing Elements (Coefficient)
• Casting Matrices
• Reshaping, Resizing, Slicing
  • Reshaping
  • Slicing
• Tensor Module

Chapter 3 Matrix Operations

• Matrix Arithmetic
  • Addition/Subtraction Matrices/ Scalar
  • Scalar Multiplication/ Division
  • Multiplication, Dot And Cross Product
  • Transposition and Conjugation
• Coefficient-Wise Operations
  • Absolute, Power, Root
  • Log, Exponential
  • Min, Max of Two Matrices
  • Finite, Inf, NaN
  • Sinusoidal
  • Floor, Ceil, Round
  • Masking Elements
• Reductions
  • Minimum/ Maximum Element In The Matrix
  • Minimum/ Maximum Element Row-wise/Col-wise in the Matrix
  • Sum, Mean, Trace, Product
  • Norms
  • All, Any, Count
  • Matrix Rank
• Matrix Condition Number and Numerical Stability
• Check Matrices Similarity
• Broadcasting

Chapter 4 Advanced Eigen Operations

• Memory Alignment
• Passing Eigen objects by value to functions
• Aliasing
• Memory Mapping
  • Eigen matrix from std::vector
• Unary Expression
• Eigen Functor

Chapter 5 Dense Linear Problems And Decompositions

• 1. Vector Space
• 2. Linear Equation
• 3. Solving Linear Equation
• 4. Matrices Decompositions
• 5. Linear Map
• 6. Span
• 7. Subspace
• 8. Range of a Matrix
• 9. Basis
• 10. Rank of Matrix
• 11. Dimension of the Column Space
• 12. Null Space (Kernel)
• 13. Nullity
• 14. Rank-nullity Theorem
• 15. The Determinant of The Matrix
• 16. Finding The Inverse of The Matrix
• 17. The Fundamental Theorem of Linear Algebra
• 18. Permutation Matrix
• 19. Augmented Matrix

Chapter 6 Sparse Matrices

• Sparse Matrix Manipulations
  • Compressed Sparse Row
• Solving Sparse Linear Systems
• Matrix Free Solvers

Chapter 7 Geometry Transformation

• 1. Euler Angles
• 2. Global References and Local Tangent Plane Coordinates
• 3. Axis-angle Representation
• 4. Quaternions
• 5. Conversion between different representations

Chapter 8 Differentiation

• Jacobian
• Hessian Matrix
• Automatic Differentiation
• Numerical Differentiation

Chapter 9 Numerical Optimization

• Newton's Method In Optimization
• Gauss-Newton Algorithm
  • Example of Gauss-Newton, Inverse Kinematic Problem
• Quasi-Newton Method
• Curve Fitting
• Non Linear Least Squares
• Non Linear Regression
• Levenberg Marquardt

Chapter 10 Linear Algebra in Robotics

• Why decompositions matter
• Inverse Kinematics — the pseudo-inverse (SVD)
• Camera Calibration — DLT (SVD), projection decomposition (QR), Zhang (Cholesky)
• SLAM — least squares, QR vs Cholesky, and sparsity
• Point-Cloud Registration — Kabsch / Umeyama (SVD)
• PCA & Plane Fitting — eigendecomposition
• Further reading & related projects