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Eigen Intro

Matrix and Vector sizes

All matrix and vector classes are subclasses of the Matrix class. There are also built-in types for commonly used matrix and vector types.

Eigen::MatrixXd A(2,4); // 2x4 double matrix
Eigen::Matrix4d B;      // 4x4 double matrix
Eigen::Matrix3f C;      // 3x3 float matrix
Eigen::VectorXf d(9);   // 9x1 float vector
Eigen::Vector4f e;      // 4x1 float vector

If you declare using namespace Eigen; beforehand in your code, you don’t need to say Eigen:: in front of these and other Eigen types. If you do this, it is highly recommended that you do this in each .cpp file and not in a .h file, where it could potentially pollute multiple other files.

Initializing a matrix

Use the << and , operators:

MatrixXd A(2,4); // 2x4 matrix
A << 1, 2, 3, 4,
     5, 6, 7, 8;

The elements are specified row by row.

You can also initialize matrices to the zero and identity matrices.

Matrix4d B = Matrix4d::Zero();     // 4x4 zero matrix
Matrix3d C = Matrix3d::Identity(); // 3x3 identity matrix
...
B.setZero();                       // back to zero
...
B.setIdentity();                   // back to identity

See http://eigen.tuxfamily.org/dox/group__TutorialAdvancedInitialization.html for more information.

Printing a matrix

You can print the matrix to the screen by using cout:

cout << A << endl;

will print out

1 2 3 4
5 6 7 8

Element access

You can access an element for read/write with the () operator. Rows and columns start with 0, not 1.

A(0,0) = 9;
double foo = A(1,3);

Arithmetic operations

Eigen overloads the common operators, so you can add, subtract, and multiply matrices easily.

Matrix4d A, B;      // 4x4 matrices
A << ...;
B << ...;
Matrix4d C = A + B; // 4x4 matrix
double s = 2.0;
Matrix4d D = s * C; // 4x4 matrix
MatrixXd E(4,3);    // 4x3 matrix
E << ...
MatrixXd F(4,3);    // 4x3 matrix
F = C * E;

The matrix sizes aren’t checked at compile time. If there is a mismatch, you’ll get a very complex looking runtime error - one drawback of template programming. Eigen warns against using the auto keyword. See http://eigen.tuxfamily.org/dox/TopicPitfalls.html. In general, it is a good idea to compile and run your code often so that Eigen errors are easier to find.

Block access

You can read/write blocks of elements with the block function.

Matrix2d B; // 2x2 matrix
B << -1, -2,
     -3, -4;
A.block<2,2>(0,0) = B;

The code above writes B into the upper 2x2 block of A. The numbers in the angled bracket indicate the size of the block, and the numbers in parenthesis indicate the starting location. You can also read a block in the same way.

Matrix2d C = A.block<2,2>(0,2); // 2x2 matrix

This sets C to be the right half of A.

For vectors, use the segment function.

Vector4d a; // 4x1 vector
a << 1, 2, 3, 4;
Vector2d b = a.segment<2>(0); // 2x1 vector

This sets b to be the first two elements of a.

Rotation matrix

You can create a rotation matrix from an axis-angle as follows. The input axis to the AngleAxis constructor must be normalized.

#include <Eigen/Geometry>
...
float angle = M_PI/6.0f;
Vector3f axis(1.0f, 1.0f, 1.0f);
axis.normalize();
Matrix3f R(AngleAxisf(angle, axis));
Matrix4f M;
M.setIdentity();
M.block<3,3>(0,0) = R;
cout << M << endl;

Similarly, you can create a rotation matrix from a unit quaternion.

#include <Eigen/Geometry>
...
float x, y, z, w;
x = y = z = w = 1.0f;
Quaternionf q(w, x, y, z);
q.normalize();
Matrix3f R(q);
Matrix4f M;
M.setIdentity();
M.block<3,3>(0,0) = R;
cout << M << endl;

Compilation Hints

• If you have structures/classes that include Eigen members, put the following inside your class: public: EIGEN_MAKE_ALIGNED_OPERATOR_NEW.
• If you’re having trouble running your application with Eigen, try #define EIGEN_DONT_ALIGN_STATICALLY before you #include <Eigen/Dense>.

For more hints, see http://eigen.tuxfamily.org/dox/group__DenseMatrixManipulation__Alignement.html.