eigen/doc/C01_QuickStartGuide.dox

namespace Eigen {

/** \page TutorialCore Tutorial 1/3 - Core features
    \ingroup Tutorial

<div class="eimainmenu">\ref index "Overview"
  | \b Core \b features
  | \ref TutorialGeometry "Geometry"
  | \ref TutorialAdvancedLinearAlgebra "Advanced linear algebra"
  | \ref TutorialSparse "Sparse matrix"
</div>

\b Table \b of \b contents
  - \ref TutorialCoreGettingStarted
  - \ref TutorialCoreSimpleExampleFixedSize
  - \ref TutorialCoreSimpleExampleDynamicSize
  - \ref TutorialCoreMatrixTypes
  - \ref TutorialCoreCoefficients
  - \ref TutorialCoreMatrixInitialization
  - \ref TutorialCoreArithmeticOperators
  - \ref TutorialCoreReductions
  - \ref TutorialCoreMatrixBlocks
  - \ref TutorialCoreDiagonalMatrices
  - \ref TutorialCoreTransposeAdjoint
  - \ref TutorialCoreDotNorm
  - \ref TutorialCoreTriangularMatrix
  - \ref TutorialCoreSelfadjointMatrix
  - \ref TutorialCoreSpecialTopics
\n

<hr>

\section TutorialCoreGettingStarted Getting started

In order to use Eigen, you just need to download and extract Eigen's source code. It is not necessary to use CMake or install anything.

Here are some quick compilation instructions with GCC. To quickly test an example program, just do

\code g++ -I /path/to/eigen2/ my_program.cpp -o my_program \endcode

There is no library to link to. For good performance, add the \c -O2 compile-flag. Note however that this makes it impossible to debug inside Eigen code, as many functions get inlined. In some cases, performance can be further improved by disabling Eigen assertions: use \c -DEIGEN_NO_DEBUG or \c -DNDEBUG to disable them.

On the x86 architecture, the SSE2 instruction set is not enabled by default. Use \c -msse2 to enable it, and Eigen will then automatically enable its vectorized paths. On x86-64 and AltiVec-based architectures, vectorization is enabled by default.

\section TutorialCoreSimpleExampleFixedSize Simple example with fixed-size matrices and vectors

By fixed-size, we mean that the number of rows and columns are fixed at compile-time. In this case, Eigen avoids dynamic memory allocation, and unroll loops when that makes sense. This is useful for very small sizes: typically up to 4x4, sometimes up to 16x16.

<table class="tutorial_code"><tr><td>
\include Tutorial_simple_example_fixed_size.cpp
</td>
<td>
output:
\include Tutorial_simple_example_fixed_size.out
</td></tr></table>

<a href="#" class="top">top</a>\section TutorialCoreSimpleExampleDynamicSize Simple example with dynamic-size matrices and vectors

By dynamic-size, we mean that the numbers of rows and columns are not fixed at compile-time. In this case, they are stored as runtime variables and the arrays are dynamically allocated.

<table class="tutorial_code"><tr><td>
\include Tutorial_simple_example_dynamic_size.cpp
</td>
<td>
output:
\include Tutorial_simple_example_dynamic_size.out
</td></tr></table>

<a name="warningarraymodule"></a>
\warning \redstar In most cases it is enough to include the \c Eigen/Core header only to get started with Eigen. However, some features presented in this tutorial require the Array module to be included (\c \#include \c <Eigen/Array>). Those features are highlighted with a red star \redstar. Notice that if you want to include all Eigen functionality at once, you can do:
\code
#include <Eigen/Eigen>
\endcode
This slows compilation down but at least you don't have to worry anymore about including the correct files! There also is the Eigen/Dense header including all dense functionality i.e. leaving out the Sparse module.


<a href="#" class="top">top</a>\section TutorialCoreMatrixTypes Matrix and vector types

In Eigen, all kinds of dense matrices and vectors are represented by the template class Matrix. In most cases, you can simply use one of the \ref matrixtypedefs "convenience typedefs".

The template class Matrix takes a number of template parameters, but for now it is enough to understand the 3 first ones (and the others can then be left unspecified):

\code Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime> \endcode

\li \c Scalar is the scalar type, i.e. the type of the coefficients. That is, if you want a vector of floats, choose \c float here.
\li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows and columns of the matrix as known at compile-time.

For example, \c Vector3d is a typedef for \code Matrix<double, 3, 1> \endcode

For dynamic-size, that is in order to left the number of rows or of columns unspecified at compile-time, use the special value Eigen::Dynamic. For example, \c VectorXd is a typedef for \code Matrix<double, Dynamic, 1> \endcode

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:

\code
Matrix<double, 6, Dynamic> // Dynamic number of columns
Matrix<double, Dynamic, 2> // Dynamic number of rows
Matrix<double, Dynamic, Dynamic> // Fully dynamic
Matrix<double, 13, 3> // Fully fixed
\endcode

Fixed-size and partially-dynamic-size matrices may use all the same API calls as fully dynamic
matrices, but the fixed dimension(s) must remain constant, or an assertion failure will occur.

<a href="#" class="top">top</a>\section TutorialCoreCoefficients Coefficient access

Eigen supports the following syntaxes for read and write coefficient access:

\code
matrix(i,j);
vector(i)
vector[i]
vector.x() // first coefficient
vector.y() // second coefficient
vector.z() // third coefficient
vector.w() // fourth coefficient
\endcode

Notice that these coefficient access methods have assertions checking the ranges. So if you do a lot of coefficient access, these assertion can have an important cost. There are then two possibilities if you want avoid paying this cost:
\li Either you can disable assertions altogether, by defining EIGEN_NO_DEBUG or NDEBUG. Notice that some IDEs like MS Visual Studio define NDEBUG automatically in "Release Mode".
\li Or you can disable the checks on a case-by-case basis by using the coeff() and coeffRef() methods: see MatrixBase::coeff(int,int) const, MatrixBase::coeffRef(int,int), etc.

<a href="#" class="top">top</a>\section TutorialCoreMatrixInitialization Matrix and vector creation and initialization

\subsection TutorialCtors Matrix constructors

The default constructor leaves coefficients uninitialized. Any dynamic size is set to 0, in which case the matrix/vector is considered uninitialized (no array is allocated).

\code
Matrix3f A; // construct 3x3 matrix with uninitialized coefficients
A(0,0) = 5; // OK
MatrixXf B; // construct 0x0 matrix without allocating anything
A(0,0) = 5; // Error, B is uninitialized, doesn't have any coefficients to address
\endcode

In the above example, B is an uninitialized matrix. What to do with such a matrix? You can call resize() on it, or you can assign another matrix to it. Like this:

\code
MatrixXf B; // uninitialized matrix
B.resize(3,5); // OK, now B is a 3x5 matrix with uninitialized coefficients
B(0,0) = 5; // OK
MatrixXf C; // uninitialized matrix
C = B; // OK, C is initialized as a copy of B
\endcode

There also are constructors taking size parameters, allowing to directly initialize dynamic-size matrices:

\code
MatrixXf B(3,5); // B is a 3x5 matrix with uninitialized coefficients
\endcode

Note that even if one of the dimensions is known at compile time, you must specify it. The only exception is vectors (i.e. matrices where one of the dimensions is known at compile time to be 1).

\code
VectorXf v(10); // OK, v is a vector of size 10 with uninitialized coefficients
Matrix<float,2,Dynamic> m(10); // Error: m is not a vector, must provide explicitly the 2 sizes
Matrix<float,2,Dynamic> m(2,10); // OK
Matrix2f m(2,2); // OK. Of course it's redundant to pass the parameters here, but it's allowed.
\endcode

For small fixed-size vectors, we also allow constructors passing the coefficients:
\code
Vector2f u(1.2f, 3.4f);
Vector3f v(1.2f, 3.4f, 5.6f);
Vector4f w(1.2f, 3.4f, 5.6f, 7.8f);
\endcode

\subsection TutorialPredefMat Predefined Matrices
Eigen offers several static methods to create special matrix expressions, and non-static methods to assign these expressions to existing matrices:

<table class="tutorial_code">
<tr>
  <td>Fixed-size matrix or vector</td>
  <td>Dynamic-size matrix</td>
  <td>Dynamic-size vector</td>
</tr>
<tr style="border-bottom-style: none;">
  <td>
\code
Matrix3f x;

x = Matrix3f::Zero();
x = Matrix3f::Ones();
x = Matrix3f::Constant(value);
x = Matrix3f::Identity();
x = Matrix3f::Random();

x.setZero();
x.setOnes();
x.setIdentity();
x.setConstant(value);
x.setRandom();
\endcode
  </td>
  <td>
\code
MatrixXf x;

x = MatrixXf::Zero(rows, cols);
x = MatrixXf::Ones(rows, cols);
x = MatrixXf::Constant(rows, cols, value);
x = MatrixXf::Identity(rows, cols);
x = MatrixXf::Random(rows, cols);

x.setZero(rows, cols);
x.setOnes(rows, cols);
x.setConstant(rows, cols, value);
x.setIdentity(rows, cols);
x.setRandom(rows, cols);
\endcode
  </td>
  <td>
\code
VectorXf x;

x = VectorXf::Zero(size);
x = VectorXf::Ones(size);
x = VectorXf::Constant(size, value);
x = VectorXf::Identity(size);
x = VectorXf::Random(size);

x.setZero(size);
x.setOnes(size);
x.setConstant(size, value);
N/A
x.setRandom(size);
\endcode
  </td>
</tr>
<tr style="border-top-style: none;"><td colspan="3">\redstar the Random() and setRandom() functions require the inclusion of the Array module (\c \#include \c <Eigen/Array>)</td></tr>
<tr><td colspan="3">Basis vectors \link MatrixBase::Unit [details]\endlink</td></tr>
<tr><td>\code
Vector3f::UnitX() // 1 0 0
Vector3f::UnitY() // 0 1 0
Vector3f::UnitZ() // 0 0 1
\endcode</td><td></td><td>\code
VectorXf::Unit(size,i)
VectorXf::Unit(4,1) == Vector4f(0,1,0,0)
                    == Vector4f::UnitY()
\endcode
</table>

Here is an usage example:
<table class="tutorial_code"><tr><td>
\code
cout << MatrixXf::Constant(2, 3, sqrt(2)) << endl;
RowVector3i v;
v.setConstant(6);
cout << "v = " << v << endl;
\endcode
</td>
<td>
output:
\code
1.41 1.41 1.41
1.41 1.41 1.41

v = 6 6 6
\endcode
</td></tr></table>


\subsection TutorialCasting Casting

In Eigen, any matrices of same size and same scalar type are all naturally compatible. The scalar type can be explicitely casted to another one using the template MatrixBase::cast() function:
\code
Matrix3d md(1,2,3);
Matrix3f mf = md.cast<float>();
\endcode
Note that casting to the same scalar type in an expression is free.

The destination matrix is automatically resized in any assignment:
\code
MatrixXf res(10,10);
Matrix3f a, b;
res = a+b; // OK: res is resized to size 3x3
\endcode
Of course, fixed-size matrices can't be resized.


\subsection TutorialMap Map
Any memory buffer can be mapped as an Eigen expression:
<table class="tutorial_code"><tr><td>
\code
std::vector<float> stlarray(10);
Map<VectorXf>(&stlarray[0], stlarray.size()).setOnes();
int data[4] = 1, 2, 3, 4;
Matrix2i mat2x2(data);
MatrixXi mat2x2 = Map<Matrix2i>(data);
MatrixXi mat2x2 = Map<MatrixXi>(data,2,2);
\endcode
</td></tr></table>



\subsection TutorialCommaInit Comma initializer
Eigen also offers a \ref MatrixBaseCommaInitRef "comma initializer syntax" which allows you to set all the coefficients of a matrix to specific values:
<table class="tutorial_code"><tr><td>
\include Tutorial_commainit_01.cpp
</td>
<td>
output:
\verbinclude Tutorial_commainit_01.out
</td></tr></table>

Not excited by the above example? Then look at the following one where the matrix is set by blocks:
<table class="tutorial_code"><tr><td>
\include Tutorial_commainit_02.cpp
</td>
<td>
output:
\verbinclude Tutorial_commainit_02.out
</td></tr></table>

<span class="note">\b Side \b note: here \link CommaInitializer::finished() .finished() \endlink
is used to get the actual matrix object once the comma initialization
of our temporary submatrix is done. Note that despite the apparent complexity of such an expression,
Eigen's comma initializer usually compiles to very optimized code without any overhead.</span>






<a href="#" class="top">top</a>\section TutorialCoreArithmeticOperators Arithmetic Operators

In short, all arithmetic operators can be used right away as in the following example. Note however that arithmetic operators are only given their usual meaning from mathematics tradition. For other operations, such as taking the coefficient-wise product of two vectors, see the discussion of \link Cwise .cwise() \endlink below. Anyway, here is an example demonstrating basic arithmetic operators:
\code
mat4 -= mat1*1.5 + mat2 * (mat3/4);
\endcode
which includes two matrix scalar products ("mat1*1.5" and "mat3/4"), a matrix-matrix product ("mat2 * (mat3/4)"),
a matrix addition ("+") and substraction with assignment ("-=").

<table class="tutorial_code">
<tr><td>
matrix/vector product</td><td>\code
col2 = mat1 * col1;
row2 = row1 * mat1;     row1 *= mat1;
mat3 = mat1 * mat2;     mat3 *= mat1; \endcode
</td></tr>
<tr><td>
add/subtract</td><td>\code
mat3 = mat1 + mat2;      mat3 += mat1;
mat3 = mat1 - mat2;      mat3 -= mat1;\endcode
</td></tr>
<tr><td>
scalar product</td><td>\code
mat3 = mat1 * s1;   mat3 = s1 * mat1;   mat3 *= s1;
mat3 = mat1 / s1;   mat3 /= s1;\endcode
</td></tr>
</table>

In Eigen, only traditional mathematical operators can be used right away.
But don't worry, thanks to the \link Cwise .cwise() \endlink operator prefix,
Eigen's matrices are also very powerful as a numerical container supporting
most common coefficient-wise operators.
<table class="noborder">
<tr><td>
<table class="tutorial_code" style="margin-right:10pt">
<tr><td>Coefficient wise \link Cwise::operator*() product \endlink</td>
<td>\code mat3 = mat1.cwise() * mat2; \endcode
</td></tr>
<tr><td>
Add a scalar to all coefficients \redstar</td><td>\code
mat3 = mat1.cwise() + scalar;
mat3.cwise() += scalar;
mat3.cwise() -= scalar;
\endcode
</td></tr>
<tr><td>
Coefficient wise \link Cwise::operator/() division \endlink \redstar</td><td>\code
mat3 = mat1.cwise() / mat2; \endcode
</td></tr>
<tr><td>
Coefficient wise \link Cwise::inverse() reciprocal \endlink \redstar</td><td>\code
mat3 = mat1.cwise().inverse(); \endcode
</td></tr>
<tr><td>
Coefficient wise comparisons \redstar \n
(support all operators)</td><td>\code
mat3 = mat1.cwise() < mat2;
mat3 = mat1.cwise() <= mat2;
mat3 = mat1.cwise() > mat2;
etc.
\endcode
</td></tr></table>
</td>
<td><table class="tutorial_code">
<tr><td>
\b Trigo \redstar: \n
\link Cwise::sin sin \endlink, \link Cwise::cos cos \endlink</td><td>\code
mat3 = mat1.cwise().sin();
etc.
\endcode
</td></tr>
<tr><td>
\b Power \redstar: \n \link Cwise::pow() pow \endlink,
\link Cwise::square square \endlink,
\link Cwise::cube cube \endlink, \n
\link Cwise::sqrt sqrt \endlink,
\link Cwise::exp exp \endlink,
\link Cwise::log log \endlink </td><td>\code
mat3 = mat1.cwise().square();
mat3 = mat1.cwise().pow(5);
mat3 = mat1.cwise().log();
etc.
\endcode
</td></tr>
<tr><td>
\link Cwise::min min \endlink, \link Cwise::max max \endlink, \n
absolute value (\link Cwise::abs() abs \endlink, \link Cwise::abs2() abs2 \endlink)
</td><td>\code
mat3 = mat1.cwise().min(mat2);
mat3 = mat1.cwise().max(mat2);
mat3 = mat1.cwise().abs();
mat3 = mat1.cwise().abs2();
\endcode</td></tr>
</table>
</td></tr></table>
\redstar Those functions require the inclusion of the Array module (\c \#include \c <Eigen/Array>).

<span class="note">\b Side \b note: If you think that the \c .cwise() syntax is too verbose for your own taste and prefer to have non-conventional mathematical operators directly available, then feel free to extend MatrixBase as described \ref ExtendingMatrixBase "here".</span>

So far, we saw the notation \code mat1*mat2 \endcode for matrix product, and \code mat1.cwise()*mat2 \endcode for coefficient-wise product. What about other kinds of products, which in some other libraries also use arithmetic operators? In Eigen, they are accessed as follows -- note that here we are anticipating on further sections, for convenience.
<table class="tutorial_code">
<tr><td>\link MatrixBase::dot() dot product \endlink (inner product)</td><td>\code
scalar = vec1.dot(vec2);\endcode
</td></tr>
<tr><td>
outer product</td><td>\code
mat = vec1 * vec2.transpose();\endcode
</td></tr>
<tr><td>
\link MatrixBase::cross() cross product \endlink</td><td>\code
#include <Eigen/Geometry>
vec3 = vec1.cross(vec2);\endcode</td></tr>
</table>




<a href="#" class="top">top</a>\section TutorialCoreReductions Reductions

Eigen provides several reduction methods such as:
\link MatrixBase::minCoeff() minCoeff() \endlink, \link MatrixBase::maxCoeff() maxCoeff() \endlink,
\link MatrixBase::sum() sum() \endlink, \link MatrixBase::trace() trace() \endlink,
\link MatrixBase::norm() norm() \endlink, \link MatrixBase::squaredNorm() squaredNorm() \endlink,
\link MatrixBase::all() all() \endlink \redstar,and \link MatrixBase::any() any() \endlink \redstar.
All reduction operations can be done matrix-wise,
\link MatrixBase::colwise() column-wise \endlink \redstar or
\link MatrixBase::rowwise() row-wise \endlink \redstar. Usage example:
<table class="tutorial_code">
<tr><td rowspan="3" style="border-right-style:dashed">\code
      5 3 1
mat = 2 7 8
      9 4 6 \endcode
</td> <td>\code mat.minCoeff(); \endcode</td><td>\code 1 \endcode</td></tr>
<tr><td>\code mat.colwise().minCoeff(); \endcode</td><td>\code 2 3 1 \endcode</td></tr>
<tr><td>\code mat.rowwise().minCoeff(); \endcode</td><td>\code
1
2
4
\endcode</td></tr>
</table>

Also note that maxCoeff and minCoeff can takes optional arguments returning the coordinates of the respective min/max coeff: \link MatrixBase::maxCoeff(int*,int*) const maxCoeff(int* i, int* j) \endlink, \link MatrixBase::minCoeff(int*,int*) const minCoeff(int* i, int* j) \endlink.

<span class="note">\b Side \b note: The all() and any() functions are especially useful in combinaison with coeff-wise comparison operators (\ref CwiseAll "example").</span>





<a href="#" class="top">top</a>\section TutorialCoreMatrixBlocks Matrix blocks

Read-write access to a \link MatrixBase::col(int) column \endlink
or a \link MatrixBase::row(int) row \endlink of a matrix:
\code
mat1.row(i) = mat2.col(j);
mat1.col(j1).swap(mat1.col(j2));
\endcode

Read-write access to sub-vectors:
<table class="tutorial_code">
<tr>
<td>Default versions</td>
<td>Optimized versions when the size \n is known at compile time</td></tr>
<td></td>

<tr><td>\code vec1.start(n)\endcode</td><td>\code vec1.start<n>()\endcode</td><td>the first \c n coeffs </td></tr>
<tr><td>\code vec1.end(n)\endcode</td><td>\code vec1.end<n>()\endcode</td><td>the last \c n coeffs </td></tr>
<tr><td>\code vec1.segment(pos,n)\endcode</td><td>\code vec1.segment<n>(pos)\endcode</td>
    <td>the \c size coeffs in \n the range [\c pos : \c pos + \c n [</td></tr>
<tr style="border-style: dashed none dashed none;"><td>

Read-write access to sub-matrices:</td><td></td><td></td></tr>
<tr>
  <td>\code mat1.block(i,j,rows,cols)\endcode
      \link MatrixBase::block(int,int,int,int) (more) \endlink</td>
  <td>\code mat1.block<rows,cols>(i,j)\endcode
      \link MatrixBase::block(int,int) (more) \endlink</td>
  <td>the \c rows x \c cols sub-matrix \n starting from position (\c i,\c j)</td></tr><tr>
 <td>\code
 mat1.corner(TopLeft,rows,cols)
 mat1.corner(TopRight,rows,cols)
 mat1.corner(BottomLeft,rows,cols)
 mat1.corner(BottomRight,rows,cols)\endcode
 \link MatrixBase::corner(CornerType,int,int) (more) \endlink</td>
 <td>\code
 mat1.corner<rows,cols>(TopLeft)
 mat1.corner<rows,cols>(TopRight)
 mat1.corner<rows,cols>(BottomLeft)
 mat1.corner<rows,cols>(BottomRight)\endcode
 \link MatrixBase::corner(CornerType) (more) \endlink</td>
 <td>the \c rows x \c cols sub-matrix \n taken in one of the four corners</td></tr>
<tr><td>\code
mat4x4.minor(i,j) = mat3x3;
mat3x3 = mat4x4.minor(i,j);\endcode
</td><td></td><td>
\link MatrixBase::minor() minor \endlink (read-write)</td>
</tr>
</table>



<a href="#" class="top">top</a>\section TutorialCoreDiagonalMatrices Diagonal matrices

<table class="tutorial_code">
<tr><td>
\link MatrixBase::asDiagonal() make a diagonal matrix \endlink from a vector \n
<em class="note">this product is automatically optimized !</em></td><td>\code
mat3 = mat1 * vec2.asDiagonal();\endcode
</td></tr>
<tr><td>Access \link MatrixBase::diagonal() the diagonal of a matrix \endlink as a vector (read/write)</td>
 <td>\code
 vec1 = mat1.diagonal();
 mat1.diagonal() = vec1;
 \endcode
</td>
</tr>
</table>

<a href="#" class="top">top</a>\section TutorialCoreTransposeAdjoint Transpose and Adjoint operations

<table class="tutorial_code">
<tr><td>
\link MatrixBase::transpose() transposition \endlink (read-write)</td><td>\code
mat3 = mat1.transpose() * mat2;
mat3.transpose() = mat1 * mat2.transpose();
\endcode
</td></tr>
<tr><td>
\link MatrixBase::adjoint() adjoint \endlink (read only)\n</td><td>\code
mat3 = mat1.adjoint() * mat2;
\endcode
</td></tr>
</table>

<a href="#" class="top">top</a>\section TutorialCoreDotNorm Dot-product, vector norm, normalization

<table class="tutorial_code">
<tr><td>
\link MatrixBase::dot() Dot-product \endlink of two vectors
</td><td>\code vec1.dot(vec2);\endcode
</td></tr>
<tr><td>
\link MatrixBase::norm() norm \endlink of a vector \n
\link MatrixBase::squaredNorm() squared norm \endlink of a vector
</td><td>\code vec.norm(); \endcode \n \code vec.squaredNorm() \endcode
</td></tr>
<tr><td>
returns a \link MatrixBase::normalized() normalized \endlink vector \n
\link MatrixBase::normalize() normalize \endlink a vector
</td><td>\code
vec3 = vec1.normalized();
vec1.normalize();\endcode
</td></tr>
</table>

<a href="#" class="top">top</a>\section TutorialCoreTriangularMatrix Dealing with triangular matrices

Currently, Eigen does not provide any explcit triangular matrix, with storage class. Instead, we
can reference a triangular part of a square matrix or expression to perform special treatment on it.
This is achieved by the class TriangularView and the MatrixBase::triangularView template function.
Note that the opposite triangular part of  the matrix is never referenced, and so it can, e.g., store
a second triangular matrix.

<table class="tutorial_code">
<tr><td>
Reference a read/write triangular part of a given \n
matrix (or expression) m with optional unit diagonal:
</td><td>\code
m.triangularView<Eigen::UpperTriangular>()
m.triangularView<Eigen::UnitUpperTriangular>()
m.triangularView<Eigen::LowerTriangular>()
m.triangularView<Eigen::UnitLowerTriangular>()\endcode
</td></tr>
<tr><td>
Writting to a specific triangular part:\n (only the referenced triangular part is evaluated)
</td><td>\code
m1.triangularView<Eigen::LowerTriangular>() = m2 + m3 \endcode
</td></tr>
<tr><td>
Convertion to a dense matrix setting the opposite triangular part to zero:
</td><td>\code
m2 = m1.triangularView<Eigen::UnitUpperTriangular>()\endcode
</td></tr>
<tr><td>
Products:
</td><td>\code
m3 += s1 * m1.adjoint().triangularView<Eigen::UnitUpperTriangular>() * m2
m3 -= s1 * m2.conjugate() * m1.adjoint().triangularView<Eigen::LowerTriangular>() \endcode
</td></tr>
<tr><td>
Solving linear equations:\n(\f$ m_2 := m_1^{-1} m_2 \f$)
</td><td>\code
m1.triangularView<Eigen::UnitLowerTriangular>().solveInPlace(m2)
m1.adjoint().triangularView<Eigen::UpperTriangular>().solveInPlace(m2)\endcode
</td></tr>
</table>


<a href="#" class="top">top</a>\section TutorialCoreSelfadjointMatrix Dealing with symmetric/selfadjoint matrices

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

<table class="tutorial_code">
<tr><td>
Conversion to a dense matrix:
</td><td>\code
m2 = m.selfadjointView<Eigen::LowerTriangular>();\endcode
</td></tr>
<tr><td>
Product with another general matrix or vector:
</td><td>\code
m3  = s1 * m1.conjugate().selfadjointView<Eigen::UpperTriangular>() * m3;
m3 -= s1 * m3.adjoint() * m1.selfadjointView<Eigen::UpperTriangular>();\endcode
</td></tr>
<tr><td>
Rank 1 and rank K update:
</td><td>\code
// fast version of m1 += s1 * m2 * m2.adjoint():
m1.selfadjointView<Eigen::UpperTriangular>().rankUpdate(m2,s1);
// fast version of m1 -= m2.adjoint() * m2:
m1.selfadjointView<Eigen::LowerTriangular>().rankUpdate(m2.adjoint(),-1); \endcode
</td></tr>
<tr><td>
Rank 2 update: (\f$ m += s u v^* + s v u^* \f$)
</td><td>\code
m.selfadjointView<Eigen::UpperTriangular>().rankUpdate(u,v,s);
\endcode
</td></tr>
<tr><td>
Solving linear equations:\n(\f$ m_2 := m_1^{-1} m_2 \f$)
</td><td>\code
// via a standard Cholesky factorization
m1.selfadjointView<Eigen::UpperTriangular>().llt().solveInPlace(m2);
// via a Cholesky factorization with pivoting
m1.selfadjointView<Eigen::UpperTriangular>().ldlt().solveInPlace(m2);
\endcode
</td></tr>
</table>


<a href="#" class="top">top</a>\section TutorialCoreSpecialTopics Special Topics

\ref TopicLazyEvaluation "Lazy Evaluation and Aliasing": Thanks to expression templates, Eigen is able to apply lazy evaluation wherever that is beneficial.

*/

}