We assume that you have already read the quick "getting started" tutorial. This page is the first one in a much longer multi-page tutorial.
Table of contents
The Matrix class takes six template parameters, but for now it's enough to learn about the first three first parameters. The three remaining parameters have default values, which for now we will leave untouched, and which we discuss below.
The three mandatory template parameters of Matrix are:
Scalaris the scalar type, i.e. the type of the coefficients. That is, if you want a matrix of floats, choose
floathere. See Scalar types for a list of all supported scalar types and for how to extend support to new types.
ColsAtCompileTimeare the number of rows and columns of the matrix as known at compile time (see below for what to do if the number is not known at compile time).
We offer a lot of convenience typedefs to cover the usual cases. For example,
Matrix4f is a 4x4 matrix of floats. Here is how it is defined by Eigen:
We discuss below these convenience typedefs.
As mentioned above, in Eigen, vectors are just a special case of matrices, with either 1 row or 1 column. The case where they have 1 column is the most common; such vectors are called column-vectors, often abbreviated as just vectors. In the other case where they have 1 row, they are called row-vectors.
For example, the convenience typedef
Vector3f is a (column) vector of 3 floats. It is defined as follows by Eigen:
We also offer convenience typedefs for row-vectors, for example:
Of course, Eigen is not limited to matrices whose dimensions are known at compile time. The
ColsAtCompileTime template parameters can take the special value
Dynamic which indicates that the size is unknown at compile time, so must be handled as a run-time variable. In Eigen terminology, such a size is referred to as a dynamic size; while a size that is known at compile time is called a fixed size. For example, the convenience typedef
MatrixXd, meaning a matrix of doubles with dynamic size, is defined as follows:
And similarly, we define a self-explanatory typedef
VectorXi as follows:
You can perfectly have e.g. a fixed number of rows with a dynamic number of columns, as in:
A default constructor is always available, never performs any dynamic memory allocation, and never initializes the matrix coefficients. You can do:
ais a 3x3 matrix, with a static float array of uninitialized coefficients,
bis a dynamic-size matrix whose size is currently 0x0, and whose array of coefficients hasn't yet been allocated at all.
Constructors taking sizes are also available. For matrices, the number of rows is always passed first. For vectors, just pass the vector size. They allocate the array of coefficients with the given size, but don't initialize the coefficients themselves:
ais a 10x15 dynamic-size matrix, with allocated but currently uninitialized coefficients.
bis a dynamic-size vector of size 30, with allocated but currently uninitialized coefficients.
In order to offer a uniform API across fixed-size and dynamic-size matrices, it is legal to use these constructors on fixed-size matrices, even if passing the sizes is useless in this case. So this is legal:
and is a no-operation.
Finally, we also offer some constructors to initialize the coefficients of small fixed-size vectors up to size 4:
The primary coefficient accessors and mutators in Eigen are the overloaded parenthesis operators. For matrices, the row index is always passed first. For vectors, just pass one index. The numbering starts at 0. This example is self-explanatory:
using namespace Eigen;
m(0,0) = 3;
m(1,0) = 2.5;
m(0,1) = -1;
m(1,1) = m(1,0) + m(0,1);
std::cout << "Here is the matrix m:\n" << m << std::endl;
v(0) = 4;
v(1) = v(0) - 1;
std::cout << "Here is the vector v:\n" << v << std::endl;
Here is the matrix m: 3 -1 2.5 1.5 Here is the vector v: 4 3
Note that the syntax
m(index) is not restricted to vectors, it is also available for general matrices, meaning index-based access in the array of coefficients. This however depends on the matrix's storage order. All Eigen matrices default to column-major storage order, but this can be changed to row-major, see Storage orders.
The operator is also overloaded for index-based access in vectors, but keep in mind that C++ doesn't allow operator to take more than one argument. We restrict operator to vectors, because an awkwardness in the C++ language would make matrix[i,j] compile to the same thing as matrix[j] !
Matrix and vector coefficients can be conveniently set using the so-called comma-initializer syntax. For now, it is enough to know this example:
m << 1, 2, 3,
4, 5, 6,
7, 8, 9;
std::cout << m;
1 2 3 4 5 6 7 8 9
The right-hand side can also contain matrix expressions as discussed in this page.
The current size of a matrix can be retrieved by rows(), cols() and size(). These methods return the number of rows, the number of columns and the number of coefficients, respectively. Resizing a dynamic-size matrix is done by the resize() method.
using namespace Eigen;
std::cout << "The matrix m is of size "
<< m.rows() << "x" << m.cols() << std::endl;
std::cout << "It has " << m.size() << " coefficients" << std::endl;
std::cout << "The vector v is of size " << v.size() << std::endl;
std::cout << "As a matrix, v is of size "
<< v.rows() << "x" << v.cols() << std::endl;
The matrix m is of size 4x3 It has 12 coefficients The vector v is of size 5 As a matrix, v is of size 5x1
The resize() method is a no-operation if the actual matrix size doesn't change; otherwise it is destructive: the values of the coefficients may change. If you want a conservative variant of resize() which does not change the coefficients, use conservativeResize(), see this page for more details.
All these methods are still available on fixed-size matrices, for the sake of API uniformity. Of course, you can't actually resize a fixed-size matrix. Trying to change a fixed size to an actually different value will trigger an assertion failure; but the following code is legal:
The matrix m is of size 4x4
Assignment is the action of copying a matrix into another, using
operator=. Eigen resizes the matrix on the left-hand side automatically so that it matches the size of the matrix on the right-hand size. For example:
a is of size 2x2 a is now of size 3x3
Of course, if the left-hand side is of fixed size, resizing it is not allowed.
If you do not want this automatic resizing to happen (for example for debugging purposes), you can disable it, see this page.
When should one use fixed sizes (e.g.
Matrix4f), and when should one prefer dynamic sizes (e.g.
MatrixXf)? The simple answer is: use fixed sizes for very small sizes where you can, and use dynamic sizes for larger sizes or where you have to. For small sizes, especially for sizes smaller than (roughly) 16, using fixed sizes is hugely beneficial to performance, as it allows Eigen to avoid dynamic memory allocation and to unroll loops. Internally, a fixed-size Eigen matrix is just a plain static array, i.e. doing
really amounts to just doing
so this really has zero runtime cost. By contrast, the array of a dynamic-size matrix is always allocated on the heap, so doing
amounts to doing
and in addition to that, the MatrixXf object stores its number of rows and columns as member variables.
The limitation of using fixed sizes, of course, is that this is only possible when you know the sizes at compile time. Also, for large enough sizes, say for sizes greater than (roughly) 32, the performance benefit of using fixed sizes becomes negligible. Worse, trying to create a very large matrix using fixed sizes could result in a stack overflow, since Eigen will try to allocate the array as a static array, which by default goes on the stack. Finally, depending on circumstances, Eigen can also be more aggressive trying to vectorize (use SIMD instructions) when dynamic sizes are used, see Vectorization.
We mentioned at the beginning of this page that the Matrix class takes six template parameters, but so far we only discussed the first three. The remaining three parameters are optional. Here is the complete list of template parameters:
Optionsis a bit field. Here, we discuss only one bit:
RowMajor. It specifies that the matrices of this type use row-major storage order; by default, the storage order is column-major. See the page on storage orders. For example, this type means row-major 3x3 matrices:
MaxColsAtCompileTimeare useful when you want to specify that, even though the exact sizes of your matrices are not known at compile time, a fixed upper bound is known at compile time. The biggest reason why you might want to do that is to avoid dynamic memory allocation. For example the following matrix type uses a static array of 12 floats, without dynamic memory allocation:
cf(meaning complex<float>), or
cd(meaning complex<double>). The fact that typedefs are only defined for these five types doesn't mean that they are the only supported scalar types. For example, all standard integer types are supported, see Scalar types.