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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.

ModuleHeader fileContents
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;

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:

MatricesArrays
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:

Basic matrix manipulation

1D objects2D objectsNotes
Constructors
Vector2f v1(x, y);
Array3i v2(x, y, z);
Vector4d v3(x, y, z, w);
VectorXf v5; // empty object
ArrayXf v6(size);
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(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

OperationCode
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.
OperationCode
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.
OperationCode
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);