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MatrixBase< Derived > Class Template Reference
Public Types | Public Member Functions | Static Public Member Functions

MatrixBase< Derived > Class Template Reference
[Core module]

Base class for all dense matrices, vectors, and expressions. More...

#include <MatrixBase.h>

Inherits DenseBase< Derived >.

Inherited by ProductBase< Derived, Lhs, Rhs >.

Public Types

typedef Matrix< Scalar,
EIGEN_SIZE_MAX(RowsAtCompileTime,
ColsAtCompileTime),
EIGEN_SIZE_MAX(RowsAtCompileTime,
ColsAtCompileTime)> 		SquareMatrixType
 type of the equivalent square matrix
typedef Matrix< typename
internal::traits< Derived >
::Scalar, internal::traits
< Derived >::RowsAtCompileTime,
internal::traits< Derived >
::ColsAtCompileTime, AutoAlign|(internal::traits
< Derived >::Flags
&RowMajorBit?RowMajor:ColMajor),
internal::traits< Derived >
::MaxRowsAtCompileTime,
internal::traits< Derived >
::MaxColsAtCompileTime > 		PlainObject
 The plain matrix type corresponding to this expression.

Public Member Functions

Index diagonalSize () const
Derived & operator= (const MatrixBase &other)
 Special case of the template operator=, in order to prevent the compiler from generating a default operator= (issue hit with g++ 4.1)
template<typename OtherDerived >
Derived & operator+= (const MatrixBase< OtherDerived > &other)
 replaces *this by *this + other.
template<typename OtherDerived >
Derived & operator-= (const MatrixBase< OtherDerived > &other)
 replaces *this by *this - other.
template<typename OtherDerived >
const ProductReturnType
< Derived, OtherDerived >
::Type 		operator* (const MatrixBase< OtherDerived > &other) const
template<typename OtherDerived >
const LazyProductReturnType
< Derived, OtherDerived >
::Type 		lazyProduct (const MatrixBase< OtherDerived > &other) const
template<typename OtherDerived >
Derived & operator*= (const EigenBase< OtherDerived > &other)
 replaces *this by *this * other.
template<typename OtherDerived >
void applyOnTheLeft (const EigenBase< OtherDerived > &other)
 replaces *this by other * *this.
template<typename OtherDerived >
void applyOnTheRight (const EigenBase< OtherDerived > &other)
 replaces *this by *this * other.
template<typename DiagonalDerived >
const DiagonalProduct< Derived,
DiagonalDerived, OnTheRight > 		operator* (const DiagonalBase< DiagonalDerived > &diagonal) const
template<typename OtherDerived >
internal::scalar_product_traits
< typename internal::traits
< Derived >::Scalar, typename
internal::traits< OtherDerived >
::Scalar >::ReturnType 		dot (const MatrixBase< OtherDerived > &other) const
template<typename OtherDerived >
Scalar eigen2_dot (const MatrixBase< OtherDerived > &other) const
RealScalar squaredNorm () const
RealScalar norm () const
RealScalar stableNorm () const
RealScalar blueNorm () const
RealScalar hypotNorm () const
const PlainObject normalized () const
void normalize ()
 Normalizes the vector, i.e.
const AdjointReturnType adjoint () const
void adjointInPlace ()
 This is the "in place" version of adjoint(): it replaces *this by its own transpose.
DiagonalReturnType diagonal ()
ConstDiagonalReturnType diagonal () const
 This is the const version of diagonal().
DiagonalDynamicIndexReturnType diagonal (Index index)
ConstDiagonalDynamicIndexReturnType diagonal (Index index) const
 This is the const version of diagonal(Index).
template<unsigned int Mode>
internal::eigen2_part_return_type
< Derived, Mode >::type 		part ()
template<unsigned int Mode>
const
internal::eigen2_part_return_type
< Derived, Mode >::type 		part () const
template<unsigned int Mode>
TriangularViewReturnType< Mode >
::Type 		triangularView ()
template<unsigned int Mode>
ConstTriangularViewReturnType
< Mode >::Type 		triangularView () const
 This is the const version of MatrixBase::triangularView()
const DiagonalWrapper< const
Derived > 		asDiagonal () const
Derived & setIdentity ()
 Writes the identity expression (not necessarily square) into *this.
Derived & setIdentity (Index rows, Index cols)
 Resizes to the given size, and writes the identity expression (not necessarily square) into *this.
bool isIdentity (const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
bool isDiagonal (const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
bool isUpperTriangular (const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
bool isLowerTriangular (const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
template<typename OtherDerived >
bool isOrthogonal (const MatrixBase< OtherDerived > &other, const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
bool isUnitary (const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
template<typename OtherDerived >
bool operator== (const MatrixBase< OtherDerived > &other) const
template<typename OtherDerived >
bool operator!= (const MatrixBase< OtherDerived > &other) const
NoAlias< Derived,
Eigen::MatrixBase > 		noalias ()
const ForceAlignedAccess< Derived > forceAlignedAccess () const
ForceAlignedAccess< Derived > forceAlignedAccess ()
template<bool Enable>
internal::add_const_on_value_type
< typename
internal::conditional< Enable,
ForceAlignedAccess< Derived >
, Derived & >::type >::type 		forceAlignedAccessIf () const
template<bool Enable>
internal::conditional< Enable,
ForceAlignedAccess< Derived >
, Derived & >::type 		forceAlignedAccessIf ()
Scalar trace () const
template<int p>
RealScalar lpNorm () const
ArrayWrapper< Derived > array ()
const FullPivLU< PlainObject > fullPivLu () const
 
const PartialPivLU< PlainObject > partialPivLu () const
 
const LU< PlainObject > lu () const
 
const internal::inverse_impl
< Derived > 		inverse () const
 
template<typename ResultType >
void computeInverseAndDetWithCheck (ResultType &inverse, typename ResultType::Scalar &determinant, bool &invertible, const RealScalar &absDeterminantThreshold=NumTraits< Scalar >::dummy_precision()) const
 
template<typename ResultType >
void computeInverseWithCheck (ResultType &inverse, bool &invertible, const RealScalar &absDeterminantThreshold=NumTraits< Scalar >::dummy_precision()) const
 
Scalar determinant () const
 
const LLT< PlainObject > llt () const
 
const LDLT< PlainObject > ldlt () const
 
const HouseholderQR< PlainObject > householderQr () const
const ColPivHouseholderQR
< PlainObject > 		colPivHouseholderQr () const
const FullPivHouseholderQR
< PlainObject > 		fullPivHouseholderQr () const
EigenvaluesReturnType eigenvalues () const
 Computes the eigenvalues of a matrix.
RealScalar operatorNorm () const
 Computes the L2 operator norm.
JacobiSVD< PlainObject > jacobiSvd (unsigned int computationOptions=0) const
 
template<typename OtherDerived >
cross_product_return_type
< OtherDerived >::type 		cross (const MatrixBase< OtherDerived > &other) const
 
template<typename OtherDerived >
PlainObject cross3 (const MatrixBase< OtherDerived > &other) const
 
PlainObject unitOrthogonal (void) const
Matrix< Scalar, 3, 1 > eulerAngles (Index a0, Index a1, Index a2) const
 
ScalarMultipleReturnType operator* (const UniformScaling< Scalar > &s) const
 Concatenates a linear transformation matrix and a uniform scaling.
HomogeneousReturnType homogeneous () const
 
const HNormalizedReturnType hnormalized () const
 
void makeHouseholderInPlace (Scalar &tau, RealScalar &beta)
 Computes the elementary reflector H such that: $ H *this = [ beta 0 ... 0]^T $ where the transformation H is: $ H = I - tau v v^*$ and the vector v is: $ v^T = [1 essential^T] $.
template<typename EssentialPart >
void makeHouseholder (EssentialPart &essential, Scalar &tau, RealScalar &beta) const
 Computes the elementary reflector H such that: $ H *this = [ beta 0 ... 0]^T $ where the transformation H is: $ H = I - tau v v^*$ and the vector v is: $ v^T = [1 essential^T] $.
template<typename EssentialPart >
void applyHouseholderOnTheLeft (const EssentialPart &essential, const Scalar &tau, Scalar *workspace)
 Apply the elementary reflector H given by $ H = I - tau v v^*$ with $ v^T = [1 essential^T] $ from the left to a vector or matrix.
template<typename EssentialPart >
void applyHouseholderOnTheRight (const EssentialPart &essential, const Scalar &tau, Scalar *workspace)
 Apply the elementary reflector H given by $ H = I - tau v v^*$ with $ v^T = [1 essential^T] $ from the right to a vector or matrix.
template<typename OtherScalar >
void applyOnTheLeft (Index p, Index q, const JacobiRotation< OtherScalar > &j)
 Applies the rotation in the plane j to the rows p and q of *this, i.e., it computes B = J * B, with $ B = \left ( \begin{array}{cc} \text{*this.row}(p) \\ \text{*this.row}(q) \end{array} \right ) $.
template<typename OtherScalar >
void applyOnTheRight (Index p, Index q, const JacobiRotation< OtherScalar > &j)
 Applies the rotation in the plane j to the columns p and q of *this, i.e., it computes B = B * J with $ B = \left ( \begin{array}{cc} \text{*this.col}(p) & \text{*this.col}(q) \end{array} \right ) $.
template<typename OtherDerived >
Derived & lazyAssign (const Flagged< OtherDerived, 0, EvalBeforeAssigningBit > &other)

Static Public Member Functions

static const IdentityReturnType Identity ()
static const IdentityReturnType Identity (Index rows, Index cols)
static const BasisReturnType Unit (Index size, Index i)
static const BasisReturnType Unit (Index i)
static const BasisReturnType UnitX ()
static const BasisReturnType UnitY ()
static const BasisReturnType UnitZ ()
static const BasisReturnType UnitW ()

Detailed Description

template<typename Derived>
class Eigen::MatrixBase< Derived >

Base class for all dense matrices, vectors, and expressions.

This class is the base that is inherited by all matrix, vector, and related expression types. Most of the Eigen API is contained in this class, and its base classes. Other important classes for the Eigen API are Matrix, and VectorwiseOp.

Note that some methods are defined in other modules such as the LU module LU module for all functions related to matrix inversions.

Template Parameters:
Derivedis the derived type, e.g. a matrix type, or an expression, etc.

When writing a function taking Eigen objects as argument, if you want your function to take as argument any matrix, vector, or expression, just let it take a MatrixBase argument. As an example, here is a function printFirstRow which, given a matrix, vector, or expression x, prints the first row of x.

    template<typename Derived>
    void printFirstRow(const Eigen::MatrixBase<Derived>& x)
    {
      cout << x.row(0) << endl;
    }

This class can be extended with the help of the plugin mechanism described on the page TopicCustomizingEigen by defining the preprocessor symbol EIGEN_MATRIXBASE_PLUGIN.

See also:
TopicClassHierarchy

Definition at line 48 of file MatrixBase.h.

Member Typedef Documentation

typedef Matrix<typename internal::traits<Derived>::Scalar, internal::traits<Derived>::RowsAtCompileTime, internal::traits<Derived>::ColsAtCompileTime, AutoAlign | (internal::traits<Derived>::Flags&RowMajorBit ? RowMajor : ColMajor), internal::traits<Derived>::MaxRowsAtCompileTime, internal::traits<Derived>::MaxColsAtCompileTime > PlainObject

The plain matrix type corresponding to this expression.

This is not necessarily exactly the return type of eval(). In the case of plain matrices, the return type of eval() is a const reference to a matrix, not a matrix! It is however guaranteed that the return type of eval() is either PlainObject or const PlainObject&.

Definition at line 115 of file MatrixBase.h.

typedef Matrix<Scalar,EIGEN_SIZE_MAX(RowsAtCompileTime,ColsAtCompileTime), EIGEN_SIZE_MAX(RowsAtCompileTime,ColsAtCompileTime)> SquareMatrixType

type of the equivalent square matrix

Definition at line 96 of file MatrixBase.h.

Member Function Documentation

const MatrixBase< Derived >::AdjointReturnType adjoint (  ) const
Returns:
an expression of the adjoint (i.e. conjugate transpose) of *this.

Example: 

 Output: 


Warning:
If you want to replace a matrix by its own adjoint, do NOT do this: 
 m = m.adjoint(); // bug!!! caused by aliasing effect

 Instead, use the adjointInPlace() method: 
 m.adjointInPlace();

 which gives Eigen good opportunities for optimization, or alternatively you can also do: 
 m = m.adjoint().eval();



See also:
adjointInPlace(), transpose(), conjugate(), class Transpose, class internal::scalar_conjugate_op 



Definition at line 237 of file Transpose.h.











      

        

          void adjointInPlace 
          (
           )
          
        

      







This is the "in place" version of adjoint(): it replaces *this by its own transpose. 


Thus, doing 


 m.adjointInPlace();

 has the same effect on m as doing 


 m = m.adjoint().eval();

 and is faster and also safer because in the latter line of code, forgetting the eval() results in a bug caused by aliasing.


Notice however that this method is only useful if you want to replace a matrix by its own adjoint. If you just need the adjoint of a matrix, use adjoint().


Note:
if the matrix is not square, then *this must be a resizable matrix. This excludes (non-square) fixed-size matrices, block-expressions and maps.


See also:
transpose(), adjoint(), transposeInPlace() 



Definition at line 323 of file Transpose.h.











      

        

          void applyHouseholderOnTheLeft 
          (
          const EssentialPart & 
           essential, 
        

        

          
          
          const Scalar & 
           tau, 
        

        

          
          
          Scalar * 
           workspace 
        

        

          
          )
          
        

      







Apply the elementary reflector H given by $ H = I - tau v v^*$ with $ v^T = [1 essential^T] $ from the left to a vector or matrix. 


On input: 


Parameters:

  

    
essentialthe essential part of the vector v 

    
tauthe scaling factor of the Householder transformation 

    
workspacea pointer to working space with at least this->cols() * essential.size() entries

  

  




See also:
MatrixBase::makeHouseholder(), MatrixBase::makeHouseholderInPlace(), MatrixBase::applyHouseholderOnTheRight() 



Definition at line 112 of file src/Householder/Householder.h.











      

        

          void applyHouseholderOnTheRight 
          (
          const EssentialPart & 
           essential, 
        

        

          
          
          const Scalar & 
           tau, 
        

        

          
          
          Scalar * 
           workspace 
        

        

          
          )
          
        

      







Apply the elementary reflector H given by $ H = I - tau v v^*$ with $ v^T = [1 essential^T] $ from the right to a vector or matrix. 


On input: 


Parameters:

  

    
essentialthe essential part of the vector v 

    
tauthe scaling factor of the Householder transformation 

    
workspacea pointer to working space with at least this->cols() * essential.size() entries

  

  




See also:
MatrixBase::makeHouseholder(), MatrixBase::makeHouseholderInPlace(), MatrixBase::applyHouseholderOnTheLeft() 



Definition at line 149 of file src/Householder/Householder.h.











      

        

          void applyOnTheLeft 
          (
          const EigenBase< OtherDerived > & 
           other )
          
        

      







replaces *this by other * *this. 


Example: 


 Output: 


 

Definition at line 556 of file MatrixBase.h.











      

        

          void applyOnTheLeft 
          (
          Index 
           p, 
        

        

          
          
          Index 
           q, 
        

        

          
          
          const JacobiRotation< OtherScalar > & 
           j 
        

        

          
          )
          
        

      







Applies the rotation in the plane j to the rows p and q of *this, i.e., it computes B = J * B, with $ B = \left ( \begin{array}{cc} \text{*this.row}(p) \\ \text{*this.row}(q) \end{array} \right ) $. 


See also:
class JacobiRotation, MatrixBase::applyOnTheRight(), internal::apply_rotation_in_the_plane() 



Definition at line 277 of file src/Jacobi/Jacobi.h.











      

        

          void applyOnTheRight 
          (
          const EigenBase< OtherDerived > & 
           other )
          
        

      







replaces *this by *this * other. 


It is equivalent to MatrixBase::operator*=().


Example: 


 Output: 


 

Definition at line 544 of file MatrixBase.h.











      

        

          ArrayWrapper<Derived> array 
          (
           )
          
        

      






Returns:
an Array  expression of this matrix 


See also:
ArrayBase::matrix() 



Definition at line 316 of file MatrixBase.h.











      

        

          const DiagonalWrapper< const Derived > asDiagonal 
          (
           )
           const
        

      






Returns:
a pseudo-expression of a diagonal matrix with *this as vector of diagonal coefficients


Example: 


 Output: 


See also:
class DiagonalWrapper, class DiagonalMatrix, diagonal(), isDiagonal() 



Definition at line 278 of file DiagonalMatrix.h.











      

        

          NumTraits< typename internal::traits< Derived >::Scalar >::Real blueNorm 
          (
           )
           const
        

      






Returns:
the l2 norm of *this using the Blue's algorithm. A Portable Fortran Program to Find the Euclidean Norm of a Vector, ACM TOMS, Vol 4, Issue 1, 1978.


For architecture/scalar types without vectorization, this version is much faster than stableNorm(). Otherwise the stableNorm() is faster.


See also:
norm(), stableNorm(), hypotNorm() 



Definition at line 184 of file StableNorm.h.











      

        

          const ColPivHouseholderQR< typename MatrixBase< Derived >::PlainObject > colPivHouseholderQr 
          (
           )
           const
        

      






Returns:
the column-pivoting Householder QR decomposition of *this.


See also:
class ColPivHouseholderQR 



Definition at line 573 of file ColPivHouseholderQR.h.











      

        

          void computeInverseAndDetWithCheck 
          (
          ResultType & 
           inverse, 
        

        

          
          
          typename ResultType::Scalar & 
           determinant, 
        

        

          
          
          bool & 
           invertible, 
        

        

          
          
          const RealScalar & 
           absDeterminantThreshold = NumTraits<Scalar>::dummy_precision() 
        

        

          
          )
           const
        

      









Computation of matrix inverse and determinant, with invertibility check.


This is only for fixed-size square matrices of size up to 4x4.


Parameters:

  

    
inverseReference to the matrix in which to store the inverse. 

    
determinantReference to the variable in which to store the determinant. 

    
invertibleReference to the bool variable in which to store whether the matrix is invertible. 

    
absDeterminantThresholdOptional parameter controlling the invertibility check. The matrix will be declared invertible if the absolute value of its determinant is greater than this threshold.

  

  




Example: 


 Output: 


See also:
inverse(), computeInverseWithCheck() 



Definition at line 347 of file Inverse.h.











      

        

          void computeInverseWithCheck 
          (
          ResultType & 
           inverse, 
        

        

          
          
          bool & 
           invertible, 
        

        

          
          
          const RealScalar & 
           absDeterminantThreshold = NumTraits<Scalar>::dummy_precision() 
        

        

          
          )
           const
        

      









Computation of matrix inverse, with invertibility check.


This is only for fixed-size square matrices of size up to 4x4.


Parameters:

  

    
inverseReference to the matrix in which to store the inverse. 

    
invertibleReference to the bool variable in which to store whether the matrix is invertible. 

    
absDeterminantThresholdOptional parameter controlling the invertibility check. The matrix will be declared invertible if the absolute value of its determinant is greater than this threshold.

  

  




Example: 


 Output: 


See also:
inverse(), computeInverseAndDetWithCheck() 



Definition at line 386 of file Inverse.h.











      

        

          MatrixBase< Derived >::template cross_product_return_type< OtherDerived >::type cross 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      









Returns:
the cross product of *this and other 


Here is a very good explanation of cross-product: http://xkcd.com/199/ 


See also:
MatrixBase::cross3() 



Definition at line 26 of file OrthoMethods.h.











      

        

          MatrixBase< Derived >::PlainObject cross3 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      









Returns:
the cross product of *this and other using only the x, y, and z coefficients


The size of *this and other must be four. This function is especially useful when using 4D vectors instead of 3D ones to get advantage of SSE/AltiVec vectorization.


See also:
MatrixBase::cross() 



Definition at line 74 of file OrthoMethods.h.











      

        

          internal::traits< Derived >::Scalar determinant 
          (
           )
           const
        

      









Returns:
the determinant of this matrix 



Definition at line 92 of file Determinant.h.











      

        

          MatrixBase< Derived >::ConstDiagonalDynamicIndexReturnType diagonal 
          (
          Index 
           index )
           const
        

      







This is the const version of diagonal(Index). 



Definition at line 202 of file Diagonal.h.











      

        

          MatrixBase< Derived >::template DiagonalIndexReturnType< Index >::Type diagonal 
          (
           )
          
        

      






Returns:
an expression of the main diagonal of the matrix *this 


*this is not required to be square.


Example: 


 Output: 


See also:
class Diagonal


Returns:
an expression of the DiagIndex-th sub or super diagonal of the matrix *this 


*this is not required to be square.


The template parameter DiagIndex represent a super diagonal if DiagIndex > 0 and a sub diagonal otherwise. DiagIndex == 0 is equivalent to the main diagonal.


Example: 


 Output: 


See also:
MatrixBase::diagonal(), class Diagonal 



Definition at line 168 of file Diagonal.h.











      

        

          MatrixBase< Derived >::template ConstDiagonalIndexReturnType< Index >::Type diagonal 
          (
           )
           const
        

      







This is the const version of diagonal(). 


This is the const version of diagonal<int>().



Definition at line 176 of file Diagonal.h.











      

        

          MatrixBase< Derived >::DiagonalDynamicIndexReturnType diagonal 
          (
          Index 
           index )
          
        

      






Returns:
an expression of the DiagIndex-th sub or super diagonal of the matrix *this 


*this is not required to be square.


The template parameter DiagIndex represent a super diagonal if DiagIndex > 0 and a sub diagonal otherwise. DiagIndex == 0 is equivalent to the main diagonal.


Example: 


 Output: 


See also:
MatrixBase::diagonal(), class Diagonal 



Definition at line 194 of file Diagonal.h.











      

        

          Index diagonalSize 
          (
           )
           const
        

      






Returns:
the size of the main diagonal, which is min(rows(),cols()). 


See also:
rows(), cols(), SizeAtCompileTime. 



Definition at line 101 of file MatrixBase.h.











      

        

          internal::scalar_product_traits< typename internal::traits< Derived >::Scalar, typename internal::traits< OtherDerived >::Scalar >::ReturnType dot 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      






Returns:
the dot product of *this with other.


Note:
If the scalar type is complex numbers, then this function returns the hermitian (sesquilinear) dot product, conjugate-linear in the first variable and linear in the second variable.


See also:
squaredNorm(), norm() 



Definition at line 63 of file Dot.h.











      

        

          internal::traits< Derived >::Scalar eigen2_dot 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      






Returns:
the dot product of *this with other, with the Eigen2 convention that the dot product is linear in the first variable (conjugating the second variable). Of course this only makes a difference in the complex case.


This method is only available in EIGEN2_SUPPORT mode.


See also:
dot() 



Definition at line 89 of file Dot.h.











      

        

          MatrixBase< Derived >::EigenvaluesReturnType eigenvalues 
          (
           )
           const
        

      







Computes the eigenvalues of a matrix. 


Returns:
Column vector containing the eigenvalues.


This function computes the eigenvalues with the help of the EigenSolver class (for real matrices) or the ComplexEigenSolver class (for complex matrices).


The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix.


The SelfAdjointView class provides a better algorithm for selfadjoint matrices.


Example: 


 Output: 


See also:
EigenSolver::eigenvalues(), ComplexEigenSolver::eigenvalues(), SelfAdjointView::eigenvalues() 



Definition at line 67 of file MatrixBaseEigenvalues.h.











      

        

          const ForceAlignedAccess< Derived > forceAlignedAccess 
          (
           )
           const
        

      






Returns:
an expression of *this with forced aligned access 


See also:
forceAlignedAccessIf(),class ForceAlignedAccess 



Definition at line 107 of file ForceAlignedAccess.h.











      

        

          ForceAlignedAccess< Derived > forceAlignedAccess 
          (
           )
          
        

      






Returns:
an expression of *this with forced aligned access 


See also:
forceAlignedAccessIf(), class ForceAlignedAccess 



Definition at line 117 of file ForceAlignedAccess.h.











      

        

          internal::add_const_on_value_type< typename internal::conditional< Enable, ForceAlignedAccess< Derived >, Derived & >::type >::type forceAlignedAccessIf 
          (
           )
           const
        

      






Returns:
an expression of *this with forced aligned access if Enable is true. 


See also:
forceAlignedAccess(), class ForceAlignedAccess 



Definition at line 128 of file ForceAlignedAccess.h.











      

        

          internal::conditional< Enable, ForceAlignedAccess< Derived >, Derived & >::type forceAlignedAccessIf 
          (
           )
          
        

      






Returns:
an expression of *this with forced aligned access if Enable is true. 


See also:
forceAlignedAccess(), class ForceAlignedAccess 



Definition at line 139 of file ForceAlignedAccess.h.











      

        

          const FullPivHouseholderQR< typename MatrixBase< Derived >::PlainObject > fullPivHouseholderQr 
          (
           )
           const
        

      






Returns:
the full-pivoting Householder QR decomposition of *this.


See also:
class FullPivHouseholderQR 



Definition at line 615 of file FullPivHouseholderQR.h.











      

        

          const FullPivLU< typename MatrixBase< Derived >::PlainObject > fullPivLu 
          (
           )
           const
        

      









Returns:
the full-pivoting LU decomposition of *this.


See also:
class FullPivLU 



Definition at line 744 of file FullPivLU.h.











      

        

          const MatrixBase< Derived >::HNormalizedReturnType hnormalized 
          (
           )
           const
        

      









Returns:
an expression of the homogeneous normalized vector of *this 


Example: 


 Output: 


See also:
VectorwiseOp::hnormalized() 



Definition at line 158 of file Homogeneous.h.











      

        

          MatrixBase< Derived >::HomogeneousReturnType homogeneous 
          (
           )
           const
        

      









Returns:
an expression of the equivalent homogeneous vector


Example: 


 Output: 


See also:
class Homogeneous 



Definition at line 127 of file Homogeneous.h.











      

        

          const HouseholderQR< typename MatrixBase< Derived >::PlainObject > householderQr 
          (
           )
           const
        

      






Returns:
the Householder QR decomposition of *this.


See also:
class HouseholderQR 



Definition at line 381 of file HouseholderQR.h.











      

        

          NumTraits< typename internal::traits< Derived >::Scalar >::Real hypotNorm 
          (
           )
           const
        

      






Returns:
the l2 norm of *this avoiding undeflow and overflow. This version use a concatenation of hypot() calls, and it is very slow.


See also:
norm(), stableNorm() 



Definition at line 196 of file StableNorm.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::IdentityReturnType Identity 
          (
           )
           [static]
        

      






Returns:
an expression of the identity matrix (not necessarily square).


This variant is only for fixed-size MatrixBase types. For dynamic-size types, you need to use the variant taking size arguments.


Example: 


 Output: 


See also:
Identity(Index,Index), setIdentity(), isIdentity() 



Definition at line 700 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::IdentityReturnType Identity 
          (
          Index 
           nbRows, 
        

        

          
          
          Index 
           nbCols 
        

        

          
          )
           [static]
        

      






Returns:
an expression of the identity matrix (not necessarily square).


The parameters nbRows and nbCols are the number of rows and of columns of the returned matrix. Must be compatible with this MatrixBase type.


This variant is meant to be used for dynamic-size matrix types. For fixed-size types, it is redundant to pass rows and cols as arguments, so Identity() should be used instead.


Example: 


 Output: 


See also:
Identity(), setIdentity(), isIdentity() 



Definition at line 683 of file CwiseNullaryOp.h.











      

        

          const internal::inverse_impl< Derived > inverse 
          (
           )
           const
        

      









Returns:
the matrix inverse of this matrix.


For small fixed sizes up to 4x4, this method uses cofactors. In the general case, this method uses class PartialPivLU.


Note:
This matrix must be invertible, otherwise the result is undefined. If you need an invertibility check, do the following: 
Example: 
 Output: 


See also:
computeInverseAndDetWithCheck() 



Definition at line 320 of file Inverse.h.











      

        

          bool isDiagonal 
          (
          const RealScalar & 
           prec = NumTraits<Scalar>::dummy_precision() )
           const
        

      






Returns:
true if *this is approximately equal to a diagonal matrix, within the precision given by prec.


Example: 


 Output: 


See also:
asDiagonal() 



Definition at line 292 of file DiagonalMatrix.h.











      

        

          bool isIdentity 
          (
          const RealScalar & 
           prec = NumTraits<Scalar>::dummy_precision() )
           const
        

      






Returns:
true if *this is approximately equal to the identity matrix (not necessarily square), within the precision given by prec.


Example: 


 Output: 


See also:
class CwiseNullaryOp, Identity(), Identity(Index,Index), setIdentity() 



Definition at line 717 of file CwiseNullaryOp.h.











      

        

          bool isLowerTriangular 
          (
          const RealScalar & 
           prec = NumTraits<Scalar>::dummy_precision() )
           const
        

      






Returns:
true if *this is approximately equal to a lower triangular matrix, within the precision given by prec.


See also:
isUpperTriangular() 



Definition at line 817 of file TriangularMatrix.h.











      

        

          bool isOrthogonal 
          (
          const MatrixBase< OtherDerived > & 
           other, 
        

        

          
          
          const RealScalar & 
           prec = NumTraits<Scalar>::dummy_precision() 
        

        

          
          )
           const
        

      






Returns:
true if *this is approximately orthogonal to other, within the precision given by prec.


Example: 


 Output: 


 

Definition at line 228 of file Dot.h.











      

        

          bool isUnitary 
          (
          const RealScalar & 
           prec = NumTraits<Scalar>::dummy_precision() )
           const
        

      






Returns:
true if *this is approximately an unitary matrix, within the precision given by prec. In the case where the Scalar type is real numbers, a unitary matrix is an orthogonal matrix, whence the name.


Note:
This can be used to check whether a family of vectors forms an orthonormal basis. Indeed, m.isUnitary() returns true if and only if the columns (equivalently, the rows) of m form an orthonormal basis.


Example: 


 Output: 


 

Definition at line 247 of file Dot.h.











      

        

          bool isUpperTriangular 
          (
          const RealScalar & 
           prec = NumTraits<Scalar>::dummy_precision() )
           const
        

      






Returns:
true if *this is approximately equal to an upper triangular matrix, within the precision given by prec.


See also:
isLowerTriangular() 



Definition at line 791 of file TriangularMatrix.h.











      

        

          JacobiSVD< typename MatrixBase< Derived >::PlainObject > jacobiSvd 
          (
          unsigned int 
           computationOptions = 0 )
           const
        

      









Returns:
the singular value decomposition of *this computed by two-sided Jacobi transformations.


See also:
class JacobiSVD 



Definition at line 969 of file JacobiSVD.h.











      

        

          Derived& lazyAssign 
          (
          const Flagged< OtherDerived, 0, EvalBeforeAssigningBit > & 
           other )
          
        

      







Definition at line 477 of file MatrixBase.h.











      

        

          const LazyProductReturnType< Derived, OtherDerived >::Type lazyProduct 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      






Returns:
an expression of the matrix product of *this and other without implicit evaluation.


The returned product will behave like any other expressions: the coefficients of the product will be computed once at a time as requested. This might be useful in some extremely rare cases when only a small and no coherent fraction of the result's coefficients have to be computed.


Warning:
This version of the matrix product can be much much slower. So use it only if you know what you are doing and that you measured a true speed improvement.


See also:
operator*(const MatrixBase&) 



Definition at line 615 of file GeneralProduct.h.











      

        

          const LDLT< typename MatrixBase< Derived >::PlainObject > ldlt 
          (
           )
           const
        

      









Returns:
the Cholesky decomposition with full pivoting without square root of *this 



Definition at line 604 of file LDLT.h.











      

        

          const LLT< typename MatrixBase< Derived >::PlainObject > llt 
          (
           )
           const
        

      









Returns:
the LLT decomposition of *this 



Definition at line 481 of file LLT.h.











      

        

          NumTraits< typename internal::traits< Derived >::Scalar >::Real lpNorm 
          (
           )
           const
        

      






Returns:
the $ \ell^p $ norm of *this, that is, returns the p-th root of the sum of the p-th powers of the absolute values of the coefficients of *this. If p is the special value Eigen::Infinity, this function returns the $ \ell^\infty $ norm, that is the maximum of the absolute values of the coefficients of *this.


See also:
norm() 



Definition at line 212 of file Dot.h.











      

        

          const PartialPivLU< typename MatrixBase< Derived >::PlainObject > lu 
          (
           )
           const
        

      









Synonym of partialPivLu().


Returns:
the partial-pivoting LU decomposition of *this.


See also:
class PartialPivLU 



Definition at line 501 of file PartialPivLU.h.











      

        

          void makeHouseholder 
          (
          EssentialPart & 
           essential, 
        

        

          
          
          Scalar & 
           tau, 
        

        

          
          
          RealScalar & 
           beta 
        

        

          
          )
           const
        

      







Computes the elementary reflector H such that: $ H *this = [ beta 0 ... 0]^T $ where the transformation H is: $ H = I - tau v v^*$ and the vector v is: $ v^T = [1 essential^T] $. 


On output: 


Parameters:

  

    
essentialthe essential part of the vector v 

    
tauthe scaling factor of the Householder transformation 

    
betathe result of H * *this 

  

  




See also:
MatrixBase::makeHouseholderInPlace(), MatrixBase::applyHouseholderOnTheLeft(), MatrixBase::applyHouseholderOnTheRight() 



Definition at line 65 of file src/Householder/Householder.h.











      

        

          void makeHouseholderInPlace 
          (
          Scalar & 
           tau, 
        

        

          
          
          RealScalar & 
           beta 
        

        

          
          )
          
        

      







Computes the elementary reflector H such that: $ H *this = [ beta 0 ... 0]^T $ where the transformation H is: $ H = I - tau v v^*$ and the vector v is: $ v^T = [1 essential^T] $. 


The essential part of the vector v is stored in *this.


On output: 


Parameters:

  

    
tauthe scaling factor of the Householder transformation 

    
betathe result of H * *this 

  

  




See also:
MatrixBase::makeHouseholder(), MatrixBase::applyHouseholderOnTheLeft(), MatrixBase::applyHouseholderOnTheRight() 



Definition at line 42 of file src/Householder/Householder.h.











      

        

          NoAlias< Derived, MatrixBase > noalias 
          (
           )
          
        

      






Returns:
a pseudo expression of *this with an operator= assuming no aliasing between *this and the source expression.


More precisely, noalias() allows to bypass the EvalBeforeAssignBit flag. Currently, even though several expressions may alias, only product expressions have this flag. Therefore, noalias() is only usefull when the source expression contains a matrix product.


Here are some examples where noalias is usefull: 


 D.noalias()  = A * B;
 D.noalias() += A.transpose() * B;
 D.noalias() -= 2 * A * B.adjoint();

On the other hand the following example will lead to a wrong result: 


 A.noalias() = A * B;

 because the result matrix A is also an operand of the matrix product. Therefore, there is no alternative than evaluating A * B in a temporary, that is the default behavior when you write: 


 A = A * B;

See also:
class NoAlias 



Definition at line 127 of file NoAlias.h.











      

        

          NumTraits< typename internal::traits< Derived >::Scalar >::Real norm 
          (
           )
           const
        

      






Returns:
, for vectors, the l2 norm of *this, and for matrices the Frobenius norm. In both cases, it consists in the square root of the sum of the square of all the matrix entries. For vectors, this is also equals to the square root of the dot product of *this with itself.


See also:
dot(), squaredNorm() 



Definition at line 125 of file Dot.h.











      

        

          void normalize 
          (
           )
          
        

      







Normalizes the vector, i.e. 


divides it by its own norm.


See also:
norm(), normalized() 



Definition at line 154 of file Dot.h.











      

        

          const MatrixBase< Derived >::PlainObject normalized 
          (
           )
           const
        

      






Returns:
an expression of the quotient of *this by its own norm.


See also:
norm(), normalize() 



Definition at line 139 of file Dot.h.











      

        

          bool operator!= 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      






Returns:
true if at least one pair of coefficients of *this and other are not exactly equal to each other. 


Warning:
When using floating point scalar values you probably should rather use a fuzzy comparison such as isApprox() 


See also:
isApprox(), operator== 



Definition at line 295 of file MatrixBase.h.











      

        

          MatrixBase< Derived >::ScalarMultipleReturnType operator* 
          (
          const UniformScaling< Scalar > & 
           s )
           const
        

      







Concatenates a linear transformation matrix and a uniform scaling. 



Definition at line 111 of file Scaling.h.











      

        

          const DiagonalProduct< Derived, DiagonalDerived, OnTheRight > operator* 
          (
          const DiagonalBase< DiagonalDerived > & 
           a_diagonal )
           const
        

      






Returns:
the diagonal matrix product of *this by the diagonal matrix diagonal. 



Definition at line 124 of file DiagonalProduct.h.











      

        

          const ProductReturnType< Derived, OtherDerived >::Type operator* 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      






Returns:
the matrix product of *this and other.


Note:
If instead of the matrix product you want the coefficient-wise product, see Cwise::operator*().


See also:
lazyProduct(), operator*=(const MatrixBase&), Cwise::operator*() 



Definition at line 574 of file GeneralProduct.h.











      

        

          Derived & operator*= 
          (
          const EigenBase< OtherDerived > & 
           other )
          
        

      







replaces *this by *this * other. 


Returns:
a reference to *this 


Example: 


 Output: 


 

Definition at line 531 of file MatrixBase.h.











      

        

          EIGEN_STRONG_INLINE Derived & operator+= 
          (
          const MatrixBase< OtherDerived > & 
           other )
          
        

      







replaces *this by *this + other. 


Returns:
a reference to *this 



Definition at line 221 of file CwiseBinaryOp.h.











      

        

          EIGEN_STRONG_INLINE Derived & operator-= 
          (
          const MatrixBase< OtherDerived > & 
           other )
          
        

      







replaces *this by *this - other. 


Returns:
a reference to *this 



Definition at line 207 of file CwiseBinaryOp.h.











      

        

          EIGEN_STRONG_INLINE Derived & operator= 
          (
          const MatrixBase< Derived > & 
           other )
          
        

      







Special case of the template operator=, in order to prevent the compiler from generating a default operator= (issue hit with g++ 4.1) 



Definition at line 562 of file Assign.h.











      

        

          bool operator== 
          (
          const MatrixBase< OtherDerived > & 
           other )
           const
        

      






Returns:
true if each coefficients of *this and other are all exactly equal. 


Warning:
When using floating point scalar values you probably should rather use a fuzzy comparison such as isApprox() 


See also:
isApprox(), operator!= 



Definition at line 287 of file MatrixBase.h.











      

        

          MatrixBase< Derived >::RealScalar operatorNorm 
          (
           )
           const
        

      







Computes the L2 operator norm. 


Returns:
Operator norm of the matrix.


This function computes the L2 operator norm of a matrix, which is also known as the spectral norm. The norm of a matrix $ A $ is defined to be 



\[ \|A\|_2 = \max_x \frac{\|Ax\|_2}{\|x\|_2} \]



 where the maximum is over all vectors and the norm on the right is the Euclidean vector norm. The norm equals the largest singular value, which is the square root of the largest eigenvalue of the positive semi-definite matrix $ A^*A $.


The current implementation uses the eigenvalues of $ A^*A $, as computed by SelfAdjointView::eigenvalues(), to compute the operator norm of a matrix. The SelfAdjointView class provides a better algorithm for selfadjoint matrices.


Example: 


 Output: 


See also:
SelfAdjointView::eigenvalues(), SelfAdjointView::operatorNorm() 



Definition at line 122 of file MatrixBaseEigenvalues.h.











      

        

          const internal::eigen2_part_return_type< Derived, Mode >::type part 
          (
           )
           const
        

      







Definition at line 743 of file TriangularMatrix.h.











      

        

          internal::eigen2_part_return_type< Derived, Mode >::type part 
          (
           )
          
        

      







Definition at line 751 of file TriangularMatrix.h.











      

        

          const PartialPivLU< typename MatrixBase< Derived >::PlainObject > partialPivLu 
          (
           )
           const
        

      









Returns:
the partial-pivoting LU decomposition of *this.


See also:
class PartialPivLU 



Definition at line 485 of file PartialPivLU.h.











      

        

          EIGEN_STRONG_INLINE Derived & setIdentity 
          (
          Index 
           nbRows, 
        

        

          
          
          Index 
           nbCols 
        

        

          
          )
          
        

      







Resizes to the given size, and writes the identity expression (not necessarily square) into *this. 


Parameters:

  

    
nbRowsthe new number of rows 

    
nbColsthe new number of columns

  

  




Example: 


 Output: 


See also:
MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Identity() 



Definition at line 788 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE Derived & setIdentity 
          (
           )
          
        

      







Writes the identity expression (not necessarily square) into *this. 


Example: 


 Output: 


See also:
class CwiseNullaryOp, Identity(), Identity(Index,Index), isIdentity() 



Definition at line 772 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE NumTraits< typename internal::traits< Derived >::Scalar >::Real squaredNorm 
          (
           )
           const
        

      






Returns:
, for vectors, the squared l2 norm of *this, and for matrices the Frobenius norm. In both cases, it consists in the sum of the square of all the matrix entries. For vectors, this is also equals to the dot product of *this with itself.


See also:
dot(), norm() 



Definition at line 113 of file Dot.h.











      

        

          NumTraits< typename internal::traits< Derived >::Scalar >::Real stableNorm 
          (
           )
           const
        

      






Returns:
the l2 norm of *this avoiding underflow and overflow. This version use a blockwise two passes algorithm: 1 - find the absolute largest coefficient s 2 - compute $ s \Vert \frac{*this}{s} \Vert $ in a standard way


For architecture/scalar types supporting vectorization, this version is faster than blueNorm(). Otherwise the blueNorm() is much faster.


See also:
norm(), blueNorm(), hypotNorm() 



Definition at line 153 of file StableNorm.h.











      

        

          EIGEN_STRONG_INLINE internal::traits< Derived >::Scalar trace 
          (
           )
           const
        

      






Returns:
the trace of *this, i.e. the sum of the coefficients on the main diagonal.


*this can be any matrix, not necessarily square.


See also:
diagonal(), sum() 



Definition at line 402 of file Redux.h.











      

        

          MatrixBase< Derived >::template ConstTriangularViewReturnType< Mode >::Type triangularView 
          (
           )
           const
        

      







This is the const version of MatrixBase::triangularView() 



Definition at line 780 of file TriangularMatrix.h.











      

        

          MatrixBase< Derived >::template TriangularViewReturnType< Mode >::Type triangularView 
          (
           )
          
        

      






Returns:
an expression of a triangular view extracted from the current matrix


The parameter Mode can have the following values: Upper, StrictlyUpper, UnitUpper, Lower, StrictlyLower, UnitLower.


Example: 


 Output: 


See also:
class TriangularView 



Definition at line 771 of file TriangularMatrix.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::BasisReturnType Unit 
          (
          Index 
           newSize, 
        

        

          
          
          Index 
           i 
        

        

          
          )
           [static]
        

      






Returns:
an expression of the i-th unit (basis) vector.


See also:
MatrixBase::Unit(Index), MatrixBase::UnitX(), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW() 



Definition at line 801 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::BasisReturnType Unit 
          (
          Index 
           i )
           [static]
        

      






Returns:
an expression of the i-th unit (basis) vector.


This variant is for fixed-size vector only.


See also:
MatrixBase::Unit(Index,Index), MatrixBase::UnitX(), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW() 



Definition at line 816 of file CwiseNullaryOp.h.











      

        

          MatrixBase< Derived >::PlainObject unitOrthogonal 
          (
          void 
           )
           const
        

      






Returns:
a unit vector which is orthogonal to *this 


The size of *this must be at least 2. If the size is exactly 2, then the returned vector is a counter clock wise rotation of *this, i.e., (-y,x).normalized().


See also:
cross() 



Definition at line 210 of file OrthoMethods.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::BasisReturnType UnitW 
          (
           )
           [static]
        

      






Returns:
an expression of the W axis unit vector (0,0,0,1)


See also:
MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW() 



Definition at line 859 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::BasisReturnType UnitX 
          (
           )
           [static]
        

      






Returns:
an expression of the X axis unit vector (1{,0}^*)


See also:
MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW() 



Definition at line 829 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::BasisReturnType UnitY 
          (
           )
           [static]
        

      






Returns:
an expression of the Y axis unit vector (0,1{,0}^*)


See also:
MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW() 



Definition at line 839 of file CwiseNullaryOp.h.











      

        

          EIGEN_STRONG_INLINE const MatrixBase< Derived >::BasisReturnType UnitZ 
          (
           )
           [static]
        

      






Returns:
an expression of the Z axis unit vector (0,0,1{,0}^*)


See also:
MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW() 



Definition at line 849 of file CwiseNullaryOp.h.









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