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authorGravatar Gael Guennebaud <g.gael@free.fr>2010-06-26 16:59:18 +0200
committerGravatar Gael Guennebaud <g.gael@free.fr>2010-06-26 16:59:18 +0200
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+namespace Eigen {
+
+/** \page QuickRefPage Quick reference guide
+
+\b Table \b of \b contents
+ - \ref QuickRef_Headers
+ - \ref QuickRef_Types
+ - \ref QuickRef_Map
+ - \ref QuickRef_ArithmeticOperators
+ - \ref QuickRef_Coeffwise
+\n
+
+<hr>
+
+<a href="#" class="top">top</a>
+\section QuickRef_Headers Modules and Header files
+
+<table class="tutorial_code">
+<tr><td>Module</td><td>Header file</td><td>Contents</td></tr>
+<tr><td>Core</td><td>\code#include <Eigen/Core>\endcode</td><td>Matrix and Array classes, basic linear algebra (including triangular and selfadjoint products), array manipulation</td></tr>
+<tr><td>Geometry</td><td>\code#include <Eigen/Geometry>\endcode</td><td>Transformation, Translation, Scaling, 2D and 3D rotations (Quaternion, AngleAxis)</td></tr>
+<tr><td>LU</td><td>\code#include <Eigen/LU>\endcode</td><td>Inverse, determinant, LU decompositions (FullPivLU, PartialPivLU) with solver</td></tr>
+<tr><td>Cholesky</td><td>\code#include <Eigen/Cholesky>\endcode</td><td>LLT and LDLT Cholesky factorization with solver</td></tr>
+<tr><td>SVD</td><td>\code#include <Eigen/SVD>\endcode</td><td>SVD decomposition with solver (HouseholderSVD, JacobiSVD)</td></tr>
+<tr><td>QR</td><td>\code#include <Eigen/QR>\endcode</td><td>QR decomposition with solver (HouseholderQR, ColPivHouseholerQR, FullPivHouseholderQR)</td></tr>
+<tr><td>Eigenvalues</td><td>\code#include <Eigen/Eigenvalues>\endcode</td><td>Eigenvalue, eigenvector decompositions for selfadjoint and non selfadjoint real or complex matrices.</td></tr>
+<tr><td>Sparse</td><td>\code#include <Eigen/Sparse>\endcode</td><td>Sparse matrix storage and related basic linear algebra.</td></tr>
+<tr><td></td><td>\code#include <Eigen/Dense>\endcode</td><td>Includes Core, Geometry, LU, Cholesky, SVD, QR, and Eigenvalues</td></tr>
+<tr><td></td><td>\code#include <Eigen/Eigen>\endcode</td><td>Includes Dense and Sparse</td></tr>
+</table>
+
+<a href="#" class="top">top</a>
+\section QuickRef_Types Array, matrix and vector types
+
+\b 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:
+\code
+typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyMatrixType;
+typedef Array<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyArrayType;
+\endcode
+
+\li \c Scalar is the scalar type of the coefficients (e.g., \c float, \c double, \c bool, \c int, etc.).
+\li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows and columns of the matrix as known at compile-time or \c Dynamic.
+\li \c Options can be \c ColMajor or \c RowMajor, default is \c ColMajor. (see class Matrix for more options)
+
+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 (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)
+\endcode
+
+In most cases, you can simply use one of the convenience typedefs for \ref matrixtypedefs "matrices" and \ref arraytypedefs "arrays". Some examples:
+<table class="tutorial_code">
+<tr><td>\code
+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
+\endcode</td><td>\code
+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
+\endcode</td></tr>
+</table>
+
+Conversion between the matrix and array worlds:
+\code
+Array44f a1, a1;
+Matrix4f m1, m2;
+m1 = a1 * a2; // OK, coeffwise product
+a1 = m1 * m2; // OK, matrix product
+a2 = a1 + m1.array();
+m2 = a1.matrix() + m1;
+ArrayWrapper<Matrix4f> m1a(m1); // m1a is an alias for m1.array(), they share the same coefficients
+MatrixWrapper<Array44f> a1m(a1);
+\endcode
+
+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:
+\li <a name="matrixonly"><a/>\matrixworld linear algebra matrix and vector only
+\li <a name="arrayonly"><a/>\arrayworld array objects only
+
+
+\subsection QuickRef_Basics Basic matrix manipulation
+
+<table class="tutorial_code">
+<tr><td></td><td>1D objects</td><td>2D objects</td><td>Notes</td></tr>
+<tr><td>Constructors</td>
+<td>\code
+Vector4d v4;
+Vector2f v1(x, y);
+Array3i v2(x, y, z);
+Vector4d v3(x, y, z, w);
+
+VectorXf v5;
+ArrayXf v6(size);
+\endcode</td><td>\code
+Matrix4f m1;
+
+
+
+
+MatrixXf m5;
+MatrixXf m6(nb_rows, nb_columns);
+\endcode</td><td>
+The coeffs are left uninitialized \n \n
+ \n \n Empty object \n The coeffs are left uninitialized</td></tr>
+<tr><td>Comma initializer</td>
+<td>\code
+Vector3f v1; v1 << x, y, z;
+ArrayXf v2(4); v2 << 1, 2, 3, 4;
+
+\endcode</td><td>\code
+Matrix3f m1; m1 << 1, 2, 3,
+ 4, 5, 6,
+ 7, 8, 9;
+\endcode</td><td></td></tr>
+
+<tr><td>Comma initializer (bis)</td>
+<td colspan="2">
+<table class="noborder"><tr><td>
+\include Tutorial_commainit_02.cpp
+</td>
+<td>
+output:
+\verbinclude Tutorial_commainit_02.out
+</td></tr></table>
+<td></td></tr>
+
+<tr><td>Runtime info</td>
+<td>\code
+vector.size();
+
+vector.innerStride();
+vector.data();
+\endcode</td><td>\code
+matrix.rows(); matrix.cols();
+matrix.innerSize(); matrix.outerSize();
+matrix.innerStride(); matrix.outerStride();
+matrix.data();
+\endcode</td><td>\n Inner/Outer* are storage order dependent</td></tr>
+<tr><td>Compile-time info</td>
+<td colspan="2">\code
+ObjectType::Scalar ObjectType::RowsAtCompileTime
+ObjectType::RealScalar ObjectType::ColsAtCompileTime
+ObjectType::Index ObjectType::SizeAtCompileTime
+\endcode</td><td></td></tr>
+<tr><td>Resizing</td>
+<td>\code
+vector.resize(size);
+
+
+vector.resizeLike(other_vector);
+vector.conservativeResize(size);
+\endcode</td><td>\code
+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);
+\endcode</td><td></td>no-op if the new sizes match,\n otherwise data are lost \n \n resizing with data preservation</tr>
+
+<tr><td>Coeff access with \n range checking</td>
+<td>\code
+vector(i) vector.x()
+vector[i] vector.y()
+ vector.z()
+ vector.w()
+\endcode</td><td>\code
+matrix(i,j)
+\endcode</td><td><em class="note">Range checking is disabled if \n NDEBUG or EIGEN_NO_DEBUG is defined</em></td></tr>
+
+<tr><td>Coeff access without \n range checking</td>
+<td>\code
+vector.coeff(i)
+vector.coeffRef(i)
+\endcode</td><td>\code
+matrix.coeff(i,j)
+matrix.coeffRef(i,j)
+\endcode</td><td></td></tr>
+
+<tr><td>Assignment/copy</td>
+<td colspan="2">\code
+object = expression;
+object_of_float = expression_of_double.cast<float>();
+\endcode</td><td>the destination is automatically resized (if possible)</td></tr>
+
+</table>
+
+\subsection QuickRef_PredefMat Predefined 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
+typedef {Matrix3f|Array33f} FixedXD;
+FixedXD x;
+
+x = FixedXD::Zero();
+x = FixedXD::Ones();
+x = FixedXD::Constant(value);
+x = FixedXD::Random();
+
+x.setZero();
+x.setOnes();
+x.setConstant(value);
+x.setRandom();
+\endcode
+ </td>
+ <td>
+\code
+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);
+
+x.setZero(rows, cols);
+x.setOnes(rows, cols);
+x.setConstant(rows, cols, value);
+x.setRandom(rows, cols);
+\endcode
+ </td>
+ <td>
+\code
+typedef {VectorXf|ArrayXf} Dynamic1D;
+Dynamic1D x;
+
+x = Dynamic1D::Zero(size);
+x = Dynamic1D::Ones(size);
+x = Dynamic1D::Constant(size, value);
+x = Dynamic1D::Random(size);
+
+x.setZero(size);
+x.setOnes(size);
+x.setConstant(size, value);
+x.setRandom(size);
+\endcode
+ </td>
+</tr>
+
+<tr><td colspan="3">Identity and \link MatrixBase::Unit basis vectors \endlink \matrixworld</td></tr>
+<tr style="border-bottom-style: none;">
+ <td>
+\code
+x = FixedXD::Identity();
+x.setIdentity();
+
+Vector3f::UnitX() // 1 0 0
+Vector3f::UnitY() // 0 1 0
+Vector3f::UnitZ() // 0 0 1
+\endcode
+ </td>
+ <td>
+\code
+x = Dynamic2D::Identity(rows, cols);
+x.setIdentity(rows, cols);
+
+
+
+N/A
+\endcode
+ </td>
+ <td>\code
+N/A
+
+
+VectorXf::Unit(size,i)
+VectorXf::Unit(4,1) == Vector4f(0,1,0,0)
+ == Vector4f::UnitY()
+\endcode
+ </td>
+</tr>
+</table>
+
+
+
+\subsection QuickRef_Map Map
+
+<table class="tutorial_code">
+<tr>
+<td>Contiguous memory</td>
+<td>\code
+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
+\endcode</td>
+</tr>
+<tr>
+<td>Typical usage of strides</td>
+<td>\code
+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<> > m1(data,2,3,OuterStride<>(3)); // == |1,4,7|
+Map<MatrixXf,0,OuterStride<3> > m2(data,2,3); // |2,5,8|
+\endcode</td>
+</tr>
+</table>
+
+
+<a href="#" class="top">top</a>
+\section QuickRef_ArithmeticOperators Arithmetic Operators
+
+<table class="tutorial_code">
+<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>
+<tr><td>
+matrix/vector product \matrixworld</td><td>\code
+col2 = mat1 * col1;
+row2 = row1 * mat1; row1 *= mat1;
+mat3 = mat1 * mat2; mat3 *= mat1; \endcode
+</td></tr>
+<tr><td>
+\link MatrixBase::dot() dot \endlink \& inner products \matrixworld</td><td>\code
+scalar = col1.adjoint() * col2;
+scalar = (col1.adjoint() * col2).value();
+scalar = vec1.dot(vec2);\endcode
+</td></tr>
+<tr><td>
+outer product \matrixworld</td><td>\code
+mat = col1 * col2.transpose();\endcode
+</td></tr>
+<tr><td>
+\link MatrixBase::cross() cross product \endlink \matrixworld</td><td>\code
+#include <Eigen/Geometry>
+vec3 = vec1.cross(vec2);\endcode</td></tr>
+</table>
+
+<a href="#" class="top">top</a>
+\section QuickRef_Coeffwise Coefficient-wise \& Array operators
+Coefficient-wise operators for matrices and vectors:
+<table class="tutorial_code">
+<tr><td>Matrix API \matrixworld</td><td>Via Array conversions</td></tr>
+<tr><td>\code
+mat1.cwiseMin(mat2)
+mat1.cwiseMax(mat2)
+mat1.cwiseAbs2()
+mat1.cwiseAbs()
+mat1.cwiseSqrt()
+mat1.cwiseProduct(mat2)
+mat1.cwiseQuotient(mat2)\endcode
+</td><td>\code
+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()
+\endcode</td></tr>
+</table>
+
+Array operators:\arrayworld
+
+<table class="tutorial_code">
+<tr><td>Arithmetic operators</td><td>\code
+array1 * array2 array1 / array2 array1 *= array2 array1 /= array2
+array1 + scalar array1 - scalar array1 += scalar array1 -= scalar
+\endcode</td></tr>
+<tr><td>Comparisons</td><td>\code
+array1 < array2 array1 > array2 array1 < scalar array1 > scalar
+array1 <= array2 array1 >= array2 array1 <= scalar array1 >= scalar
+array1 == array2 array1 != array2 array1 == scalar array1 != scalar
+\endcode</td></tr>
+<tr><td>Special functions \n and STL variants</td><td>\code
+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)
+\endcode
+</td></tr>
+</table>
+
+*/
+
+// FIXME I stopped here
+
+/**
+<a href="#" class="top">top</a>
+\section TutorialCoreReductions Reductions
+
+Eigen provides several reduction methods such as:
+\link DenseBase::minCoeff() minCoeff() \endlink, \link DenseBase::maxCoeff() maxCoeff() \endlink,
+\link DenseBase::sum() sum() \endlink, \link MatrixBase::trace() trace() \endlink \matrixworld,
+\link MatrixBase::norm() norm() \endlink \matrixworld, \link MatrixBase::squaredNorm() squaredNorm() \endlink \matrixworld,
+\link DenseBase::all() all() \endlink \redstar,and \link DenseBase::any() any() \endlink \redstar.
+All reduction operations can be done matrix-wise,
+\link DenseBase::colwise() column-wise \endlink \redstar or
+\link DenseBase::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 DenseBase::maxCoeff(int*,int*) const maxCoeff(int* i, int* j) \endlink, \link DenseBase::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 combination with coeff-wise comparison operators.</span>
+
+
+
+
+
+<a href="#" class="top">top</a>\section TutorialCoreMatrixBlocks Matrix blocks
+
+Read-write access to a \link DenseBase::col(int) column \endlink
+or a \link DenseBase::row(int) row \endlink of a matrix (or array):
+\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.head(n)\endcode</td><td>\code vec1.head<n>()\endcode</td><td>the first \c n coeffs </td></tr>
+<tr><td>\code vec1.tail(n)\endcode</td><td>\code vec1.tail<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 DenseBase::block(int,int,int,int) (more) \endlink</td>
+ <td>\code mat1.block<rows,cols>(i,j)\endcode
+ \link DenseBase::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.topLeftCorner(rows,cols)
+ mat1.topRightCorner(rows,cols)
+ mat1.bottomLeftCorner(rows,cols)
+ mat1.bottomRightCorner(rows,cols)\endcode
+ <td>\code
+ mat1.topLeftCorner<rows,cols>()
+ mat1.topRightCorner<rows,cols>()
+ mat1.bottomLeftCorner<rows,cols>()
+ mat1.bottomRightCorner<rows,cols>()\endcode
+ <td>the \c rows x \c cols sub-matrix \n taken in one of the four corners</td></tr>
+</table>
+
+
+
+<a href="#" class="top">top</a>\section TutorialCoreDiagonalMatrices Diagonal matrices
+\matrixworld
+
+<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 DenseBase::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) \matrixworld\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 \matrixworld
+
+<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 \matrixworld
+
+Currently, Eigen does not provide any explicit 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>
+Writing 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>
+Conversion 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$)
+</td><td>\code
+m1.triangularView<Eigen::UnitLowerTriangular>().solveInPlace(m2)
+m1.adjoint().triangularView<Eigen::UpperTriangular>().solveInPlace(m2)\endcode
+</td></tr>
+</table>
+
+
+
+<a href="#" class="top">top</a>
+\section TutorialCoreSelfadjointMatrix Dealing with symmetric/selfadjoint matrices \matrixworld
+
+Just as for triangular matrix, you can reference any triangular part of a square matrix to see it a selfadjoint
+matrix to perform special and optimized operations. Again the opposite triangular is never referenced and can be
+used to store other information.
+
+<table class="tutorial_code">
+<tr><td>
+Conversion to a dense matrix:
+</td><td>\code
+m2 = m.selfadjointView<Eigen::LowerTriangular>();\endcode
+</td></tr>
+<tr><td>
+Product with another general matrix or vector:
+</td><td>\code
+m3 = s1 * m1.conjugate().selfadjointView<Eigen::UpperTriangular>() * m3;
+m3 -= s1 * m3.adjoint() * m1.selfadjointView<Eigen::UpperTriangular>();\endcode
+</td></tr>
+<tr><td>
+Rank 1 and rank K update:
+</td><td>\code
+// fast version of m1 += s1 * m2 * m2.adjoint():
+m1.selfadjointView<Eigen::UpperTriangular>().rankUpdate(m2,s1);
+// fast version of m1 -= m2.adjoint() * m2:
+m1.selfadjointView<Eigen::LowerTriangular>().rankUpdate(m2.adjoint(),-1); \endcode
+</td></tr>
+<tr><td>
+Rank 2 update: (\f$ 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.
+
+*/
+
+}