Eigen
3.3.9

This page discusses several advanced methods for initializing matrices. It gives more details on the commainitializer, which was introduced before. It also explains how to get special matrices such as the identity matrix and the zero matrix.
Eigen offers a comma initializer syntax which allows the user to easily set all the coefficients of a matrix, vector or array. Simply list the coefficients, starting at the topleft corner and moving from left to right and from the top to the bottom. The size of the object needs to be specified beforehand. If you list too few or too many coefficients, Eigen will complain.
Example:  Output: 

Matrix3f m;
m << 1, 2, 3,
4, 5, 6,
7, 8, 9;
std::cout << m;
 1 2 3 4 5 6 7 8 9 
Moreover, the elements of the initialization list may themselves be vectors or matrices. A common use is to join vectors or matrices together. For example, here is how to join two row vectors together. Remember that you have to set the size before you can use the comma initializer.
Example:  Output: 

RowVectorXd vec1(3);
vec1 << 1, 2, 3;
std::cout << "vec1 = " << vec1 << std::endl;
RowVectorXd vec2(4);
vec2 << 1, 4, 9, 16;
std::cout << "vec2 = " << vec2 << std::endl;
RowVectorXd joined(7);
joined << vec1, vec2;
std::cout << "joined = " << joined << std::endl;
 vec1 = 1 2 3 vec2 = 1 4 9 16 joined = 1 2 3 1 4 9 16 
We can use the same technique to initialize matrices with a block structure.
Example:  Output: 

MatrixXf matA(2, 2);
matA << 1, 2, 3, 4;
MatrixXf matB(4, 4);
matB << matA, matA/10, matA/10, matA;
std::cout << matB << std::endl;
 1 2 0.1 0.2 3 4 0.3 0.4 0.1 0.2 1 2 0.3 0.4 3 4 
The comma initializer can also be used to fill block expressions such as m.row(i)
. Here is a more complicated way to get the same result as in the first example above:
Example:  Output: 

Matrix3f m;
m.row(0) << 1, 2, 3;
m.block(1,0,2,2) << 4, 5, 7, 8;
m.col(2).tail(2) << 6, 9;
std::cout << m;
 1 2 3 4 5 6 7 8 9 
The Matrix and Array classes have static methods like Zero(), which can be used to initialize all coefficients to zero. There are three variants. The first variant takes no arguments and can only be used for fixedsize objects. If you want to initialize a dynamicsize object to zero, you need to specify the size. Thus, the second variant requires one argument and can be used for onedimensional dynamicsize objects, while the third variant requires two arguments and can be used for twodimensional objects. All three variants are illustrated in the following example:
Example:  Output: 

std::cout << "A fixedsize array:\n";
Array33f a1 = Array33f::Zero();
std::cout << a1 << "\n\n";
std::cout << "A onedimensional dynamicsize array:\n";
ArrayXf a2 = ArrayXf::Zero(3);
std::cout << a2 << "\n\n";
std::cout << "A twodimensional dynamicsize array:\n";
ArrayXXf a3 = ArrayXXf::Zero(3, 4);
std::cout << a3 << "\n";
 A fixedsize array: 0 0 0 0 0 0 0 0 0 A onedimensional dynamicsize array: 0 0 0 A twodimensional dynamicsize array: 0 0 0 0 0 0 0 0 0 0 0 0 
Similarly, the static method Constant(value) sets all coefficients to value
. If the size of the object needs to be specified, the additional arguments go before the value
argument, as in MatrixXd::Constant(rows, cols, value)
. The method Random() fills the matrix or array with random coefficients. The identity matrix can be obtained by calling Identity(); this method is only available for Matrix, not for Array, because "identity matrix" is a linear algebra concept. The method LinSpaced(size, low, high) is only available for vectors and onedimensional arrays; it yields a vector of the specified size whose coefficients are equally spaced between low
and high
. The method LinSpaced()
is illustrated in the following example, which prints a table with angles in degrees, the corresponding angle in radians, and their sine and cosine.
Example:  Output: 

ArrayXXf table(10, 4);
table.col(0) = ArrayXf::LinSpaced(10, 0, 90);
table.col(1) = M_PI / 180 * table.col(0);
table.col(2) = table.col(1).sin();
table.col(3) = table.col(1).cos();
std::cout << " Degrees Radians Sine Cosine\n";
std::cout << table << std::endl;
 Degrees Radians Sine Cosine 0 0 0 1 10 0.175 0.174 0.985 20 0.349 0.342 0.94 30 0.524 0.5 0.866 40 0.698 0.643 0.766 50 0.873 0.766 0.643 60 1.05 0.866 0.5 70 1.22 0.94 0.342 80 1.4 0.985 0.174 90 1.57 1 4.37e08 
This example shows that objects like the ones returned by LinSpaced() can be assigned to variables (and expressions). Eigen defines utility functions like setZero(), MatrixBase::setIdentity() and DenseBase::setLinSpaced() to do this conveniently. The following example contrasts three ways to construct the matrix \( J = \bigl[ \begin{smallmatrix} O & I \\ I & O \end{smallmatrix} \bigr] \): using static methods and assignment, using static methods and the commainitializer, or using the setXxx() methods.
Example:  Output: 

const int size = 6;
MatrixXd mat1(size, size);
mat1.topLeftCorner(size/2, size/2) = MatrixXd::Zero(size/2, size/2);
mat1.topRightCorner(size/2, size/2) = MatrixXd::Identity(size/2, size/2);
mat1.bottomLeftCorner(size/2, size/2) = MatrixXd::Identity(size/2, size/2);
mat1.bottomRightCorner(size/2, size/2) = MatrixXd::Zero(size/2, size/2);
std::cout << mat1 << std::endl << std::endl;
MatrixXd mat2(size, size);
mat2.topLeftCorner(size/2, size/2).setZero();
mat2.topRightCorner(size/2, size/2).setIdentity();
mat2.bottomLeftCorner(size/2, size/2).setIdentity();
mat2.bottomRightCorner(size/2, size/2).setZero();
std::cout << mat2 << std::endl << std::endl;
MatrixXd mat3(size, size);
mat3 << MatrixXd::Zero(size/2, size/2), MatrixXd::Identity(size/2, size/2),
MatrixXd::Identity(size/2, size/2), MatrixXd::Zero(size/2, size/2);
std::cout << mat3 << std::endl;
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 
A summary of all predefined matrix, vector and array objects can be found in the Quick reference guide.
As shown above, static methods as Zero() and Constant() can be used to initialize variables at the time of declaration or at the righthand side of an assignment operator. You can think of these methods as returning a matrix or array; in fact, they return socalled expression objects which evaluate to a matrix or array when needed, so that this syntax does not incur any overhead.
These expressions can also be used as a temporary object. The second example in the Getting started guide, which we reproduce here, already illustrates this.
Example:  Output: 

#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
using namespace std;
int main()
{
MatrixXd m = MatrixXd::Random(3,3);
m = (m + MatrixXd::Constant(3,3,1.2)) * 50;
cout << "m =" << endl << m << endl;
VectorXd v(3);
v << 1, 2, 3;
cout << "m * v =" << endl << m * v << endl;
}
 m = 10 55.9 14.7 23.2 63.3 77.9 85.6 31.9 77.9 m * v = 166 383 383 
The expression m + MatrixXf::Constant(3,3,1.2)
constructs the 3by3 matrix expression with all its coefficients equal to 1.2 plus the corresponding coefficient of m.
The commainitializer, too, can also be used to construct temporary objects. The following example constructs a random matrix of size 2by3, and then multiplies this matrix on the left with \( \bigl[ \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \bigr] \).
Example:  Output: 

MatrixXf mat = MatrixXf::Random(2, 3);
std::cout << mat << std::endl << std::endl;
mat = (MatrixXf(2,2) << 0, 1, 1, 0).finished() * mat;
std::cout << mat << std::endl;
 1 0.511 0.0655 0.737 0.0827 0.562 0.737 0.0827 0.562 1 0.511 0.0655 
The finished() method is necessary here to get the actual matrix object once the comma initialization of our temporary submatrix is done.