eigen/doc/C01_QuickStartGuide.dox

namespace Eigen {

/** \page TutorialCore Tutorial 1/4 - 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/eigen/ 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. To turn off Eigen's vectorization use \c-DEIGEN_DONT_VECTORIZE.

\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 Array, matrix and vector types

Eigen provides two kinds of dense objects: mathematical matrices and vectors which are both represented by the template class Matrix, and 1D and 2D arrays represented by the template class Array. While the former (Matrix) is specialized for the representation of mathematical objects, the latter (Array) represents a collection of scalar values arranged in a 1D or 2D fashion. As a major difference, all operations performed on arrays are coefficient wise. Matrix and Array have a lot of similarities since they both inherits the DenseBase and DenseStorageBase classes. In the rest of this tutorial 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 for matrix/vector only features
\li <a name="arrayonly"><a/>\arrayworld for array only features

Note that conversion between the two worlds can be done using the MatrixBase::array() and ArrayBase::matrix() functions respectively without any overhead.

In most cases, you can simply use one of the convenience typedefs for \ref matrixtypedefs "matrices" and \ref arraytypedefs "arrays".

The template class Matrix (just like the class Array) take 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>
Array<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.

Finally, note that the default typedefs for array containers is slighlty different as we have to distinghish between 1D and 2D arrays:
\code
ArrayXf     // 1D dynamic array of floats
Array2i     // 1D array of integers of size 2
ArrayXXd    // 2D fully dynamic array of doubles
Array44f    // 2D array of floats of size 4x4
\endcode


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

Eigen supports the following syntaxes for read and write coefficient access of matrices, vectors and arrays:

\code
matrix(i,j);
vector(i)
vector[i]
\endcode
Vectors support also the following additional read-write accessors:
\code
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 DenseBase::coeff(int,int) const, DenseBase::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
B(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.
The following are

<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::Identity();
x = FixedXD::Random();

x.setZero();
x.setOnes();
x.setIdentity();
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::Identity(rows, cols);
x = Dynamic2D::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
typedef {VectorXf|ArrayXf} Dynamic1D;
Dynamic1D x;

x = Dynamic1D::Zero(size);
x = Dynamic1D::Ones(size);
x = Dynamic1D::Constant(size, value);
x = Dynamic1D::Identity(size);
x = Dynamic1D::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">The following are for matrix only: \matrixworld</td></tr>
<tr style="border-bottom-style: none;">
  <td>
\code
x = FixedXD::Identity();
x.setIdentity();
\endcode
  </td>
  <td>
\code
x = Dynamic2D::Identity(rows, cols);
x.setIdentity(rows, cols);
\endcode
  </td>
  <td>
  </td>
</tr>
<tr><td colspan="3">Basis vectors \matrixworld \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 explicitly casted to another one using the template DenseBase::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.

An array object or expression can be directly assigned to a matrix, and vice versa:
\code
Matrix4f res;
Array44f a, b;
res = a * b;
\endcode
On the other hand, an array and a matrix expressions cannot be mixed in an expression, and one have to be converted to the other using the MatrixBase::array() \matrixworld and ArrayBase::matrix() \arrayworld functions respectively:
\code
Matrix4f m1, m2;
Array44f a1, a2;
m2 = a1 * m1.array();  // coeffwise product
a2 = a1.matrix() * m1; // matrix product
\endcode
Finally it is possible to declare a variable wrapping a matrix as an array object and vice versa:
\code
MatrixXf m1;
ArrayWrapper<MatrixXf> a1(m1);  // a1 and m1 share the same coefficients
// now you can use a1 as an alias for m1.array()
ArrayXXf a2;
MatrixWrapper<ArrayXXf> m2(a1); // a2 and m2 share the same coefficients
// ...
\endcode

\subsection TutorialMap Map
Any memory buffer can be mapped as an Eigen expression using the Map() static method:
\code
std::vector<float> stlarray(10);
VectorXf::Map(&stlarray[0], stlarray.size()).squaredNorm();
\endcode
Here VectorXf::Map returns an object of class Map<VectorXf>, which behaves like a VectorXf except that it uses the existing array. You can write to this object, that will write to the existing array. You can also construct a named obtect to reuse it:
\code
float data[rows*cols];
Map<MatrixXf> m(data,rows,cols);
m = othermatrix1 * othermatrix2;
m.eigenvalues();
\endcode
In the fixed-size case, no need to pass sizes:
\code
float data[9];
Map<Matrix3d> m(data);
Matrix3d::Map(data).setIdentity();
\endcode


\subsection TutorialCommaInit Comma initializer
Eigen also offers a \ref MatrixBaseCommaInitRef "comma initializer syntax" which allows you to set all the coefficients of any dense objects (matrix, vector, array, block, etc.) 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 for matrices and vectors arithmetic operators are only given their usual meaning from mathematics tradition while all array operators are performed coefficient wise.

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 subtraction with assignment ("-=").

<table class="tutorial_code">
<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>
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>
Other coefficient wise operators</td><td>\code
mat1.cwiseProduct(mat2);  mat1.cwiseQuotient(mat2);
mat1.cwiseMin(mat2);      mat1.cwiseMax(mat2);
mat1.cwiseAbs2();         mat1.cwiseSqrt();
mat1.cwiseAbs();\endcode
</td></tr>
</table>

In addition to the above operators, array objects supports all kind of coefficient wise operators which usually apply to scalar values. Recall that those operators can be used on matrices by converting them to arrays using the array() function (see \ref TutorialCasting Casting).
<table class="noborder">
<tr><td>
<table class="tutorial_code" style="margin-right:10pt">
<tr><td>Coefficient wise \link ArrayBase::operator*() product \arrayworld \endlink</td>
<td>\code array3 = array1 * array2; \endcode
</td></tr>
<tr><td>
Add a scalar to all coefficients</td><td>\code
array3 = array1 + scalar;
array3 += scalar;
array3 -= scalar;
\endcode
</td></tr>
<tr><td>
Coefficient wise \link ArrayBase::operator/() division \endlink \arrayworld</td><td>\code
array3 = array1 / array2; \endcode
</td></tr>
<tr><td>
Coefficient wise \link ArrayBase::inverse() reciprocal \endlink \arrayworld</td><td>\code
array3 = array1.inverse(); \endcode
</td></tr>
<tr><td>
Coefficient wise comparisons \arrayworld \n
(support all operators)</td><td>\code
array3 = array1 < array2;
array3 = array1 <= array2;
array3 = array1 > array2;
etc.
\endcode
</td></tr></table>
</td>
<td><table class="tutorial_code">
<tr><td>
\b Trigo \arrayworld: \n
\link ArrayBase::sin sin \endlink, \link ArrayBase::cos cos \endlink</td><td>\code
array3 = array1.sin();
etc.
\endcode
</td></tr>
<tr><td>
\b Power \arrayworld: \n \link ArrayBase::pow() pow \endlink,
\link ArrayBase::square square \endlink,
\link ArrayBase::cube cube \endlink, \n
\link ArrayBase::sqrt sqrt \endlink,
\link ArrayBase::exp exp \endlink,
\link ArrayBase::log log \endlink </td><td>\code
array3 = array1.square();
array3 = array1.pow(5);
array3 = array1.log();
etc.
\endcode
</td></tr>
<tr><td>
\link ArrayBase::min min \endlink, \link ArrayBase::max max \endlink, \n
absolute value (\link ArrayBase::abs() abs \endlink, \link ArrayBase::abs2() abs2 \endlink \arrayworld)
</td><td>\code
array3 = array1.min(array2);
array3 = array1.max(array2);
array3 = array1.abs();
array3 = array1.abs2();
\endcode</td></tr>
</table>
</td></tr></table>

So far, we saw the notation \code mat1*mat2 \endcode for matrix product, and \code array1*array2 \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) \matrixworld</td><td>\code
scalar = vec1.dot(vec2);\endcode
</td></tr>
<tr><td>
outer product \matrixworld</td><td>\code
mat = vec1 * vec2.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 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.corner(TopLeft,rows,cols)
 mat1.corner(TopRight,rows,cols)
 mat1.corner(BottomLeft,rows,cols)
 mat1.corner(BottomRight,rows,cols)\endcode
 \link DenseBase::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 DenseBase::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 DenseBase::minor() minor \endlink (read-write)</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>



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\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>



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\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>


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\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.

*/

}