[SLAM Notes] How to Use Eigen for Matrix Operations

SLAM Notes series:https://blog.csdn.net/weixin_44543463/category_10925276.html

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I. Introduction to and Installation of the Eigen Library

1.1 What Is Eigen?

  Eigen is an open-source C++ linear algebra library that provides fast matrix and linear algebra operations, equation solving, and other features. Many higher-level software libraries also use Eigen for matrix operations.   Eigen is a header-only library, so to use it, you only need to include its header files; no library linking is required.

1.2 Installing Eigen

  If Eigen is not installed on your computer, enter the following command to install it.

sudo apt-get install libeigen3-dev

II. Basic Use of the Eigen Library

2.1 Including Eigen

Add the following two header files at the beginning of the cpp file

#include <Eigen/Core>
#include <Eigen/Dense>

Specify Eigen’s header-file directory in CMakeLists.txt (if you installed Eigen in a different location, you must change the header-file directory here)

include_directories("/usr/include/eigen3")

2.2 Basic Eigen Syntax

Function	Syntax
Declare an m-by-n float matrix	eigen Eigen::Matrix<float,m,n> matrix_name;
Declare a three-dimensional column vector	Eigen::Vector3d vector_name;
Declare a three-by-three square matrix	Eigen::Matrix3d matrix_name;
Dynamic-size matrix	Eigen::Matrix<double,Eigen::Dynamic,Eigen::Dynamic> matrix_name;
Initialize a matrix as a zero matrix	Eigen::Matrix3d matrix_name = Eigen::Matrix3d::Zero();
Input data	matrix_name << 1,2,3,4,5,6;
Output data	cout << matrix_name << endl;
Data type conversion	matrix_name.cast<double>()
Matrix multiplication	matrix_name1 * matrix_name2
Transpose	matrix_name.transpose();
Sum of all elements	matrix_name.sum()
Trace	matrix_name.trace()
Inverse	matrix_name.inverse()
Determinant	matrix_name.determinant()
Conjugate matrix	matrix_name.conjugate()
Adjoint matrix	matrix_adjoint()
Compute eigenvalues	Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> eigenSolver(matrix3d);
Eigenvalues	eigenSolver.eigenvalues()
Eigenvectors	eigenSolver.eigenvectors()

2.3 Implementing Rotation Transformations with Eigen

1. Three-dimensional rotation matrix: Simply create a three-dimensional matrix

//例：创建一个三阶单位矩阵
Eigen::Matrix3d rotation_matrix = Eigen::Matrix3d::Identity();

2. Rotation vector: Use AngleAxis. You can multiply it by a vector to perform a rotation (because the operator is overloaded). The arguments in parentheses are the rotation vector’s angle and axis

//创建一个绕z轴旋转45°的旋转向量
Eigen::AngleAxisd rotation_vector(M_PI/4,Eigen::Vector3d(0,0,1));

3. Convert a rotation vector to a three-dimensional rotation matrix

//将rotation_vector这个旋转向量转换为旋转矩阵并打印出来
cout<<"rotation matrix = \n"<<rotation_vector.matrix()<<endl;
//或通过toRotationMatrix转换为旋转矩阵
rotation_matrix = rotation_vector.toRotationMatrix();

4. Convert a rotation matrix to Euler angles

//将旋转矩阵转换为ZYX顺序的欧拉角，即yaw-pitch-roll
Eigen::Vector3d euler_angles = rotation_matrix.eulerAngles(2,1,0);
cout<<"yaw pitch roll = "<<euler_angles.transpose()<<endl;

5. Perform a coordinate transformation with a rotation vector: Because the operator is overloaded, you can perform the rotation simply by multiplying the rotation vector by the coordinate vector

//定义一个x方向的向量，用前面定义的旋转向量进行旋转，然后输出旋转后的结果。
Eigen::Vector3d v(1,0,0);
Eigen::Vector3d v_rotated = rotation_vector * v;
cout<<"(1,0,0) after rotation = "<<v_rotated.transpose()<<endl;

6. Perform a coordinate transformation with a rotation matrix

v_rotated = rotation_matrix * v;
cout<<"(1,0,0) after rotation = "<<v_rotated.transpose()<<endl;

7. Perform a coordinate transformation with a transformation matrix

//定义一个名为T的变换矩阵,虽说是3d，但实际是4x4矩阵，Identity说明旋转是0，平移也是0
Eigen::Isometry3d T=Eigen::Isometry3d::Identity();
//将左上角的旋转矩阵设为按旋转向量rotation_vector旋转
T.rotate(rotation_vector);
//设置右上角的平移矩阵为[1,3,4]（旋转前平移）
T.pretranslate(Eigen::Vector3d(1,3,4));
//因为运算符重载，变换矩阵可以直接乘三维向量
Eigen::Vector3d v_transformed = T*v;

8. Use a quaternion

//创建一个四元数
Eigen::Quaterniond q;
//可以直接把旋转向量赋值给四元数
q = Eigen::Quaterniond(rotation_vector);
//也可以把旋转矩阵赋值给它
q=Eigen::Quaterniond(rotation_matrix);
//使用四元数旋转一个向量
v_rotated = q*v;