Quick reference guide

Table of contents

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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 (static allocation)

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 
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: 

  





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)            std::min(array1,array2)
array1.max(array2)            std::max(array1,array2)
array1.abs2()
array1.abs()                  std::abs(array1)
array1.sqrt()                 std::sqrt(array1)
array1.log()                  std::log(array1)
array1.exp()                  std::exp(array1)
array1.pow(exponent)          std::pow(array1,exponent)
array1.square()
array1.cube()
array1.inverse()
array1.sin()                  std::sin(array1)
array1.cos()                  std::cos(array1)
array1.tan()                  std::tan(array1)
array1.asin()                 std::asin(array1)
array1.acos()                 std::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 [ 






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.





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.





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);

  










re_vision

Author(s): Dorian Galvez-Lopez

autogenerated on Sun Jan 5 2014 11:33:51