Modules and Header files

The Eigen library is divided in a Core module and several additional modules. Each module has a corresponding header file which has to be included in order to use the module. The Dense and Eigen header files are provided to conveniently gain access to several modules at once.

Module	Header file	Contents
Core	#include <Eigen/Core>	Matrix and Array classes, basic linear algebra (including triangular and selfadjoint products), array manipulation
Geometry	#include <Eigen/Geometry>	Transform, Translation, Scaling, Rotation2D and 3D rotations (Quaternion, AngleAxis)
LU	#include <Eigen/LU>	Inverse, determinant, LU decompositions with solver (FullPivLU, PartialPivLU)
Cholesky	#include <Eigen/Cholesky>	LLT and LDLT Cholesky factorization with solver
Householder	#include <Eigen/Householder>	Householder transformations; this module is used by several linear algebra modules
SVD	#include <Eigen/SVD>	SVD decomposition with least-squares solver (JacobiSVD)
QR	#include <Eigen/QR>	QR decomposition with solver (HouseholderQR, ColPivHouseholderQR, FullPivHouseholderQR)
Eigenvalues	#include <Eigen/Eigenvalues>	Eigenvalue, eigenvector decompositions (EigenSolver, SelfAdjointEigenSolver, ComplexEigenSolver)
Sparse	#include <Eigen/Sparse>	Sparse matrix storage and related basic linear algebra (SparseMatrix, DynamicSparseMatrix, SparseVector)
	#include <Eigen/Dense>	Includes Core, Geometry, LU, Cholesky, SVD, QR, and Eigenvalues header files
	#include <Eigen/Eigen>	Includes Dense and Sparse header files (the whole Eigen library)

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Array, matrix and vector types

Recall: Eigen provides two kinds of dense objects: mathematical matrices and vectors which are both represented by the template class Matrix, and general 1D and 2D arrays represented by the template class Array:

typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyMatrixType;

typedef Array<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyArrayType;

Scalar is the scalar type of the coefficients (e.g., float, double, bool, int, etc.).
RowsAtCompileTime and ColsAtCompileTime are the number of rows and columns of the matrix as known at compile-time or Dynamic.
Options can be ColMajor or RowMajor, default is ColMajor. (see class Matrix for more options)

All combinations are allowed: you can have a matrix with a fixed number of rows and a dynamic number of columns, etc. The following are all valid:

Matrix<double, 6, Dynamic>                  // Dynamic number of columns (heap allocation)
Matrix<double, Dynamic, 2>                  // Dynamic number of rows (heap allocation)
Matrix<double, Dynamic, Dynamic, RowMajor>  // Fully dynamic, row major (heap allocation)
Matrix<double, 13, 3>                       // Fully fixed (usually allocated on stack)

In most cases, you can simply use one of the convenience typedefs for matrices and arrays. Some examples:

Matrices	Arrays
Matrix<float,Dynamic,Dynamic> <=> MatrixXf Matrix<double,Dynamic,1> <=> VectorXd Matrix<int,1,Dynamic> <=> RowVectorXi Matrix<float,3,3> <=> Matrix3f Matrix<float,4,1> <=> Vector4f	Array<float,Dynamic,Dynamic> <=> ArrayXXf Array<double,Dynamic,1> <=> ArrayXd Array<int,1,Dynamic> <=> RowArrayXi Array<float,3,3> <=> Array33f Array<float,4,1> <=> Array4f

Conversion between the matrix and array worlds:

Array44f a1, a1;
Matrix4f m1, m2;
m1 = a1 * a2;                     // coeffwise product, implicit conversion from array to matrix.
a1 = m1 * m2;                     // matrix product, implicit conversion from matrix to array.
a2 = a1 + m1.array();             // mixing array and matrix is forbidden
m2 = a1.matrix() + m1;            // and explicit conversion is required.
ArrayWrapper<Matrix4f> m1a(m1);   // m1a is an alias for m1.array(), they share the same coefficients
MatrixWrapper<Array44f> a1m(a1);

In the rest of this document we will use the following symbols to emphasize the features which are specifics to a given kind of object:

* linear algebra matrix and vector only
* array objects only

Basic matrix manipulation

	1D objects	2D objects	Notes
Constructors	Vector4d v4; Vector2f v1(x, y); Array3i v2(x, y, z); Vector4d v3(x, y, z, w); VectorXf v5; // empty object ArrayXf v6(size);	Matrix4f m1; MatrixXf m5; // empty object MatrixXf m6(nb_rows, nb_columns);	By default, the coefficients are left uninitialized
Comma initializer	Vector3f v1; v1 << x, y, z; ArrayXf v2(4); v2 << 1, 2, 3, 4;	Matrix3f m1; m1 << 1, 2, 3, 4, 5, 6, 7, 8, 9;
Comma initializer (bis)	int rows=5, cols=5; MatrixXf m(rows,cols); m << (Matrix3f() << 1, 2, 3, 4, 5, 6, 7, 8, 9).finished(), MatrixXf::Zero(3,cols-3), MatrixXf::Zero(rows-3,3), MatrixXf::Identity(rows-3,cols-3); cout << m;		output: 1 2 3 0 0 4 5 6 0 0 7 8 9 0 0 0 0 0 1 0 0 0 0 0 1
Runtime info	vector.size(); vector.innerStride(); vector.data();	matrix.rows(); matrix.cols(); matrix.innerSize(); matrix.outerSize(); matrix.innerStride(); matrix.outerStride(); matrix.data();	Inner/Outer* are storage order dependent
Compile-time info	ObjectType::Scalar ObjectType::RowsAtCompileTime ObjectType::RealScalar ObjectType::ColsAtCompileTime ObjectType::Index ObjectType::SizeAtCompileTime
Resizing	vector.resize(size); vector.resizeLike(other_vector); vector.conservativeResize(size);	matrix.resize(nb_rows, nb_cols); matrix.resize(Eigen::NoChange, nb_cols); matrix.resize(nb_rows, Eigen::NoChange); matrix.resizeLike(other_matrix); matrix.conservativeResize(nb_rows, nb_cols);	no-op if the new sizes match, otherwise data are lost resizing with data preservation
Coeff access with range checking	vector(i) vector.x() vector[i] vector.y() vector.z() vector.w()	matrix(i,j)	Range checking is disabled if NDEBUG or EIGEN_NO_DEBUG is defined
Coeff access without range checking	vector.coeff(i) vector.coeffRef(i)	matrix.coeff(i,j) matrix.coeffRef(i,j)
Assignment/copy	object = expression; object_of_float = expression_of_double.cast<float>();		the destination is automatically resized (if possible)

Predefined Matrices

Fixed-size matrix or vector	Dynamic-size matrix	Dynamic-size vector
typedef {Matrix3f\|Array33f} FixedXD; FixedXD x; x = FixedXD::Zero(); x = FixedXD::Ones(); x = FixedXD::Constant(value); x = FixedXD::Random(); x = FixedXD::LinSpaced(size, low, high); x.setZero(); x.setOnes(); x.setConstant(value); x.setRandom(); x.setLinSpaced(size, low, high);	typedef {MatrixXf\|ArrayXXf} Dynamic2D; Dynamic2D x; x = Dynamic2D::Zero(rows, cols); x = Dynamic2D::Ones(rows, cols); x = Dynamic2D::Constant(rows, cols, value); x = Dynamic2D::Random(rows, cols); N/A x.setZero(rows, cols); x.setOnes(rows, cols); x.setConstant(rows, cols, value); x.setRandom(rows, cols); N/A	typedef {VectorXf\|ArrayXf} Dynamic1D; Dynamic1D x; x = Dynamic1D::Zero(size); x = Dynamic1D::Ones(size); x = Dynamic1D::Constant(size, value); x = Dynamic1D::Random(size); x = Dynamic1D::LinSpaced(size, low, high); x.setZero(size); x.setOnes(size); x.setConstant(size, value); x.setRandom(size); x.setLinSpaced(size, low, high);
Identity and basis vectors *
x = FixedXD::Identity(); x.setIdentity(); Vector3f::UnitX() // 1 0 0 Vector3f::UnitY() // 0 1 0 Vector3f::UnitZ() // 0 0 1	x = Dynamic2D::Identity(rows, cols); x.setIdentity(rows, cols); N/A	N/A VectorXf::Unit(size,i) VectorXf::Unit(4,1) == Vector4f(0,1,0,0) == Vector4f::UnitY()

Mapping external arrays

Contiguous memory	float data[] = {1,2,3,4}; Map<Vector3f> v1(data); // uses v1 as a Vector3f object Map<ArrayXf> v2(data,3); // uses v2 as a ArrayXf object Map<Array22f> m1(data); // uses m1 as a Array22f object Map<MatrixXf> m2(data,2,2); // uses m2 as a MatrixXf object
Typical usage of strides	float data[] = {1,2,3,4,5,6,7,8,9}; Map<VectorXf,0,InnerStride<2> > v1(data,3); // = [1,3,5] Map<VectorXf,0,InnerStride<> > v2(data,3,InnerStride<>(3)); // = [1,4,7] Map<MatrixXf,0,OuterStride<3> > m2(data,2,3); // both lines \|1,4,7\| Map<MatrixXf,0,OuterStride<> > m1(data,2,3,OuterStride<>(3)); // are equal to: \|2,5,8\|

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Arithmetic Operators

add subtract	mat3 = mat1 + mat2; mat3 += mat1; mat3 = mat1 - mat2; mat3 -= mat1;
scalar product	mat3 = mat1 * s1; mat3 = s1; mat3 = s1 mat1; mat3 = mat1 / s1; mat3 /= s1;
matrix/vector products *	col2 = mat1 * col1; row2 = row1 * mat1; row1 = mat1; mat3 = mat1 mat2; mat3 *= mat1;
transposition adjoint *	mat1 = mat2.transpose(); mat1.transposeInPlace(); mat1 = mat2.adjoint(); mat1.adjointInPlace();
dot product inner product *	scalar = vec1.dot(vec2); scalar = col1.adjoint() * col2; scalar = (col1.adjoint() * col2).value();
outer product *	mat = col1 * col2.transpose();
norm normalization *	scalar = vec1.norm(); scalar = vec1.squaredNorm() vec2 = vec1.normalized(); vec1.normalize(); // inplace
cross product *	#include <Eigen/Geometry> vec3 = vec1.cross(vec2);

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Coefficient-wise & Array operators

Coefficient-wise operators for matrices and vectors:

Matrix API *	Via Array conversions
mat1.cwiseMin(mat2) mat1.cwiseMax(mat2) mat1.cwiseAbs2() mat1.cwiseAbs() mat1.cwiseSqrt() mat1.cwiseProduct(mat2) mat1.cwiseQuotient(mat2)	mat1.array().min(mat2.array()) mat1.array().max(mat2.array()) mat1.array().abs2() mat1.array().abs() mat1.array().sqrt() mat1.array() * mat2.array() mat1.array() / mat2.array()

It is also very simple to apply any user defined function foo using DenseBase::unaryExpr together with std::ptr_fun:

mat1.unaryExpr(std::ptr_fun(foo))

Array operators:*

Arithmetic operators	array1 * array2 array1 / array2 array1 *= array2 array1 /= array2 array1 + scalar array1 - scalar array1 += scalar array1 -= scalar
Comparisons	array1 < array2 array1 > array2 array1 < scalar array1 > scalar array1 <= array2 array1 >= array2 array1 <= scalar array1 >= scalar array1 == array2 array1 != array2 array1 == scalar array1 != scalar
Trigo, power, and misc functions and the STL variants	array1.min(array2) array1.max(array2) array1.abs2() array1.abs() abs(array1) array1.sqrt() sqrt(array1) array1.log() log(array1) array1.exp() exp(array1) array1.pow(exponent) pow(array1,exponent) array1.square() array1.cube() array1.inverse() array1.sin() sin(array1) array1.cos() cos(array1) array1.tan() tan(array1) array1.asin() asin(array1) array1.acos() acos(array1)

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Reductions

Eigen provides several reduction methods such as: minCoeff() , maxCoeff() , sum() , prod() , trace() *, norm() *, squaredNorm() *, all() , and any() . All reduction operations can be done matrix-wise, column-wise or row-wise . Usage example:

5 3 1 mat = 2 7 8 9 4 6	mat.minCoeff();	1
	mat.colwise().minCoeff();	2 3 1
	mat.rowwise().minCoeff();	1 2 4

Special versions of minCoeff and maxCoeff :

int i, j;
s = vector.minCoeff(&i);        // s == vector[i]
s = matrix.maxCoeff(&i, &j);    // s == matrix(i,j)

Typical use cases of all() and any():

if((array1 > 0).all()) ... // if all coefficients of array1 are greater than 0 ...

if((array1 < array2).any()) ... // if there exist a pair i,j such that array1(i,j) < array2(i,j) ...

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Sub-matrices

Read-write access to a column or a row of a matrix (or array):

mat1.row(i) = mat2.col(j);

mat1.col(j1).swap(mat1.col(j2));

Read-write access to sub-vectors:

Default versions	Optimized versions when the size is known at compile time
vec1.head(n)	vec1.head<n>()	the first `n` coeffs
vec1.tail(n)	vec1.tail<n>()	the last `n` coeffs
vec1.segment(pos,n)	vec1.segment<n>(pos)	the `n` coeffs in the range [`pos` : `pos` + `n` - 1]
Read-write access to sub-matrices:
mat1.block(i,j,rows,cols) (more)	mat1.block<rows,cols>(i,j) (more)	the `rows` x `cols` sub-matrix starting from position (`i`,`j`)
mat1.topLeftCorner(rows,cols) mat1.topRightCorner(rows,cols) mat1.bottomLeftCorner(rows,cols) mat1.bottomRightCorner(rows,cols)	mat1.topLeftCorner<rows,cols>() mat1.topRightCorner<rows,cols>() mat1.bottomLeftCorner<rows,cols>() mat1.bottomRightCorner<rows,cols>()	the `rows` x `cols` sub-matrix taken in one of the four corners
mat1.topRows(rows) mat1.bottomRows(rows) mat1.leftCols(cols) mat1.rightCols(cols)	mat1.topRows<rows>() mat1.bottomRows<rows>() mat1.leftCols<cols>() mat1.rightCols<cols>()	specialized versions of block() when the block fit two corners

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Miscellaneous operations

Reverse

Vectors, rows, and/or columns of a matrix can be reversed (see DenseBase::reverse(), DenseBase::reverseInPlace(), VectorwiseOp::reverse()).

vec.reverse() mat.colwise().reverse() mat.rowwise().reverse()

vec.reverseInPlace()

Replicate

Vectors, matrices, rows, and/or columns can be replicated in any direction (see DenseBase::replicate(), VectorwiseOp::replicate())

vec.replicate(times)                                          vec.replicate<Times>
mat.replicate(vertical_times, horizontal_times)               mat.replicate<VerticalTimes, HorizontalTimes>()
mat.colwise().replicate(vertical_times, horizontal_times)     mat.colwise().replicate<VerticalTimes, HorizontalTimes>()
mat.rowwise().replicate(vertical_times, horizontal_times)     mat.rowwise().replicate<VerticalTimes, HorizontalTimes>()

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Diagonal, Triangular, and Self-adjoint matrices

(matrix world *)

Diagonal matrices

Operation	Code
view a vector as a diagonal matrix	mat1 = vec1.asDiagonal();
Declare a diagonal matrix	DiagonalMatrix<Scalar,SizeAtCompileTime> diag1(size); diag1.diagonal() = vector;
Access the diagonal and super/sub diagonals of a matrix as a vector (read/write)	vec1 = mat1.diagonal(); mat1.diagonal() = vec1; // main diagonal vec1 = mat1.diagonal(+n); mat1.diagonal(+n) = vec1; // n-th super diagonal vec1 = mat1.diagonal(-n); mat1.diagonal(-n) = vec1; // n-th sub diagonal vec1 = mat1.diagonal<1>(); mat1.diagonal<1>() = vec1; // first super diagonal vec1 = mat1.diagonal<-2>(); mat1.diagonal<-2>() = vec1; // second sub diagonal
Optimized products and inverse	mat3 = scalar * diag1 * mat1; mat3 += scalar * mat1 * vec1.asDiagonal(); mat3 = vec1.asDiagonal().inverse() * mat1 mat3 = mat1 * diag1.inverse()

Triangular views

TriangularView gives a view on a triangular part of a dense matrix and allows to perform optimized operations on it. The opposite triangular part is never referenced and can be used to store other information.

Note: The .triangularView() template member function requires the template keyword if it is used on an object of a type that depends on a template parameter; see The template and typename keywords in C++ for details.

Operation	Code
Reference to a triangular with optional unit or null diagonal (read/write):	m.triangularView<Xxx>() `Xxx` = Upper, Lower, StrictlyUpper, StrictlyLower, UnitUpper, UnitLower
Writing to a specific triangular part: (only the referenced triangular part is evaluated)	m1.triangularView<Eigen::Lower>() = m2 + m3
Conversion to a dense matrix setting the opposite triangular part to zero:	m2 = m1.triangularView<Eigen::UnitUpper>()
Products:	m3 += s1 * m1.adjoint().triangularView<Eigen::UnitUpper>() * m2 m3 -= s1 * m2.conjugate() * m1.adjoint().triangularView<Eigen::Lower>()
Solving linear equations: $M_2 := L_1^{-1} M_2$ $M_3 := {L_1^*}^{-1} M_3$ $M_4 := M_4 U_1^{-1}$	L1.triangularView<Eigen::UnitLower>().solveInPlace(M2) L1.triangularView<Eigen::Lower>().adjoint().solveInPlace(M3) U1.triangularView<Eigen::Upper>().solveInPlace<OnTheRight>(M4)

Symmetric/selfadjoint views

Just as for triangular matrix, you can reference any triangular part of a square matrix to see it as a selfadjoint matrix and perform special and optimized operations. Again the opposite triangular part is never referenced and can be used to store other information.

Note: The .selfadjointView() template member function requires the template keyword if it is used on an object of a type that depends on a template parameter; see The template and typename keywords in C++ for details.

Operation	Code
Conversion to a dense matrix:	m2 = m.selfadjointView<Eigen::Lower>();
Product with another general matrix or vector:	m3 = s1 * m1.conjugate().selfadjointView<Eigen::Upper>() * m3; m3 -= s1 * m3.adjoint() * m1.selfadjointView<Eigen::Lower>();
Rank 1 and rank K update: $upper(M_1) \mathrel{{+}{=}} s_1 M_2 M_2^$ $lower(M_1) \mathbin{{-}{=}} M_2^ M_2$	M1.selfadjointView<Eigen::Upper>().rankUpdate(M2,s1); M1.selfadjointView<Eigen::Lower>().rankUpdate(M2.adjoint(),-1);
Rank 2 update: ( $M \mathrel{{+}{=}} s u v^* + s v u^*$ )	M.selfadjointView<Eigen::Upper>().rankUpdate(u,v,s);
Solving linear equations: ( $M_2 := M_1^{-1} M_2$ )	// via a standard Cholesky factorization m2 = m1.selfadjointView<Eigen::Upper>().llt().solve(m2); // via a Cholesky factorization with pivoting m2 = m1.selfadjointView<Eigen::Lower>().ldlt().solve(m2);