Collection of Linear Algebra functions. More...

#include <LinearAlgebra.hpp>

Classes
struct	EigenMatrix
	Type for Eigen matrices used to reduce namespace cluttering. More...

struct	EigenVector
	Type for Eigen vectors, used to reduce namespace cluttering. More...

Static Public Member Functions
static void	svd (const Eigen::Matrix< double, -1, -1 > &M, Eigen::Matrix< double, -1, -1 > &U, Eigen::Matrix< double, -1, 1 > &sigma, Eigen::Matrix< double, -1, -1 > &V)
	Performs a singular value decomposition (SVD) More...

static Eigen::Matrix< double, -1, -1 >	pseudoInverse (const Eigen::Matrix< double, -1, -1 > &am, double precision=1e-6)
	Calculates the moore-penrose (pseudo) inverse of a matrix \( \mathbf{M}^+\). More...

static bool	checkPenroseConditions (const Eigen::Matrix< double, -1, -1 > &A, const Eigen::Matrix< double, -1, -1 > &X, double prec)
	Checks the penrose conditions. More...

template<class R >
static double	det (const Eigen::MatrixBase< R > &m)
	Calculates matrix determinant. More...

template<class T >
static T	inverse (const Eigen::MatrixBase< T > &M)
	Calculates matrix inverse. More...

template<class R >
static bool	isSO (const Eigen::MatrixBase< R > &M)
	Checks if a given matrix is in SO(n) (special orthogonal) More...

template<class R >
static bool	isSO (const Eigen::MatrixBase< R > &M, typename R::Scalar precision)
	Checks if a given matrix is in SO(n) (special orthogonal) More...

template<class R >
static bool	isSkewSymmetric (const Eigen::MatrixBase< R > &M)
	Checks if a given matrix is skew-symmetrical. More...

template<class R >
static bool	isProperOrthonormal (const Eigen::MatrixBase< R > &r, typename R::Scalar precision=std::numeric_limits< typename R::Scalar >::epsilon())
	Checks if a given matrix is proper orthonormal. More...

template<class R >
static bool	isOrthonormal (const Eigen::MatrixBase< R > &r, typename R::Scalar precision=std::numeric_limits< typename R::Scalar >::epsilon())
	Checks if a given matrix is orthonormal. More...

template<class T >
static std::pair< typename EigenMatrix< T >::type, typename EigenVector< T >::type >	eigenDecompositionSymmetric (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &Am1)
	Decomposition for a symmetric matrix. More...

template<class T >
static std::pair< typename EigenMatrix< std::complex< T > >::type, typename EigenVector< std::complex< T > >::type >	eigenDecomposition (const typename Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &Am1)
	Eigen decomposition of a matrix. More...

template<>
std::pair< typename LinearAlgebra::EigenMatrix< double >::type, typename LinearAlgebra::EigenVector< double >::type >	eigenDecompositionSymmetric (const Eigen::MatrixXd &Am1)

template<>
std::pair< typename LinearAlgebra::EigenMatrix< std::complex< double > >::type, typename LinearAlgebra::EigenVector< std::complex< double > >::type >	eigenDecomposition (const Eigen::MatrixXd &Am1)

Detailed Description

Collection of Linear Algebra functions.

Member Function Documentation

◆ checkPenroseConditions()

static bool checkPenroseConditions	(	const Eigen::Matrix< double, -1, -1 > &	A,
		const Eigen::Matrix< double, -1, -1 > &	X,
		double	prec
	)

static

Checks the penrose conditions.

Parameters

A	[in] a matrix
X	[in] a pseudoinverse of A
prec	[in] the tolerance

Returns: true if the pseudoinverse X of A fullfills the penrose conditions, false otherwise

Checks the penrose conditions:

\( AXA = A \)

\( XAX = X \)

\( (AX)^T = AX \)

\( (XA)^T = XA \)

◆ det()

static double det ( const Eigen::MatrixBase< R > & m )

inlinestatic

Calculates matrix determinant.

Parameters

m	[in] a square matrix

Returns: the matrix determinant

◆ eigenDecomposition()

static std::pair<typename EigenMatrix<std::complex<T> >::type, typename EigenVector<std::complex<T> >::type> eigenDecomposition ( const typename Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > & Am1 )

inlinestatic

Eigen decomposition of a matrix.

Parameters

Am1	[in] the matrix.

Returns: the decomposition as a pair with eigenvectors and eigenvalues.

◆ eigenDecompositionSymmetric()

static std::pair<typename EigenMatrix<T>::type, typename EigenVector<T>::type> eigenDecompositionSymmetric ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > & Am1 )

inlinestatic

Decomposition for a symmetric matrix.

Parameters

Am1	[in] a symmetric matrix.

Returns: the decomposition as a pair with eigenvectors and eigenvalues.

◆ inverse()

static T inverse ( const Eigen::MatrixBase< T > & M )

inlinestatic

Calculates matrix inverse.

Parameters

M	[in] input matrix \( \mathbf{M} \) to invert

Returns: output matrix \( \mathbf{M}^{-1} \)

◆ isOrthonormal()

static bool isOrthonormal	(	const Eigen::MatrixBase< R > &	r,
		typename R::Scalar	precision = `std::numeric_limits<typename R::Scalar>::epsilon()`
	)

inlinestatic

Checks if a given matrix is orthonormal.

Returns: true if the matrix is orthonormal, false otherwise

A matrix is orthonormal if all of it's column's are mutually orthogonal and all of it's column's has unit length.

that is for any \( i, j \) the following holds \( col_i . col_j = 0 \) and \( ||col_i|| = 1 \)

Another nessesary and sufficient condition of orthonormal matrices is that \( \mathbf{M}\mathbf{M}^T=I \)

◆ isProperOrthonormal()

static bool isProperOrthonormal	(	const Eigen::MatrixBase< R > &	r,
		typename R::Scalar	precision = `std::numeric_limits<typename R::Scalar>::epsilon()`
	)

inlinestatic

Checks if a given matrix is proper orthonormal.

Returns: true if the matrix is proper orthonormal, false otherwise

A matrix is proper orthonormal if it is orthonormal and its determinant is equal to \( +1 \)

◆ isSkewSymmetric()

static bool isSkewSymmetric ( const Eigen::MatrixBase< R > & M )

inlinestatic

Checks if a given matrix is skew-symmetrical.

Parameters

M	[in] \( \mathbf{M} \) the matrix to check

Returns: true if the property \( \mathbf{M}=-\mathbf{M}^T \) holds, false otherwise.

◆ isSO() [1/2]

static bool isSO ( const Eigen::MatrixBase< R > & M )

inlinestatic

Checks if a given matrix is in SO(n) (special orthogonal)

Parameters

M	[in] \( \mathbf{M} \)

Returns: true if \( M\in SO(n) \)

\( SO(n) = {\mathbf{R}\in \mathbb{R}^{n\times n} : \mathbf{R}\mathbf{R}^T=\mathbf{I}, det \mathbf{R}=+1} \)

◆ isSO() [2/2]

static bool isSO	(	const Eigen::MatrixBase< R > &	M,
		typename R::Scalar	precision
	)

inlinestatic

Checks if a given matrix is in SO(n) (special orthogonal)

Parameters

M	[in] \( \mathbf{M} \)
precision	[in] the precision to use for floating point comparison

Returns: true if \( M\in SO(n) \)

\( SO(n) = {\mathbf{R}\in \mathbb{R}^{n\times n} : \mathbf{R}\mathbf{R}^T=\mathbf{I}, det \mathbf{R}=+1} \)

◆ pseudoInverse()

static Eigen::Matrix<double, -1, -1> pseudoInverse	(	const Eigen::Matrix< double, -1, -1 > &	am,
		double	precision = `1e-6`
	)

static

Calculates the moore-penrose (pseudo) inverse of a matrix \( \mathbf{M}^+\).

Parameters

am	[in] the matrix \( \mathbf{M} \) to be inverted
precision	[in] the precision to use, values below this treshold are considered singular

Returns: the pseudo-inverse \( \mathbf{M}^+\) of \( \mathbf{M} \)

\( \mathbf{M}^+=\mathbf{V}\mathbf{\Sigma} ^+\mathbf{U}^T \) where \( \mathbf{V} \), \( \mathbf{\Sigma} \) and \( \mathbf{U} \) are optained using Singular Value Decomposition (SVD)

◆ svd()

static void svd	(	const Eigen::Matrix< double, -1, -1 > &	M,
		Eigen::Matrix< double, -1, -1 > &	U,
		Eigen::Matrix< double, -1, 1 > &	sigma,
		Eigen::Matrix< double, -1, -1 > &	V
	)

static

Performs a singular value decomposition (SVD)

The SVD computes the decomposition \( \mathbf{M}=\mathbf{U}*\mathbf{DiagonalMatrix(\sigma)}*\mathbf{V}^T \) .

Parameters

M	[in] the matrix to decomposite
U	[out] Result matrix \(\mathbf{U}\)
sigma	[out] The \(\mathbf{sigma}\) vector with diagonal elements
V	[out] Result matrix \(\mathbf{V}\)

The documentation for this class was generated from the following file:

LinearAlgebra.hpp

Classes

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ checkPenroseConditions()

◆ det()

◆ eigenDecomposition()

◆ eigenDecompositionSymmetric()

◆ inverse()

◆ isOrthonormal()

◆ isProperOrthonormal()

◆ isSkewSymmetric()

◆ isSO() [1/2]

◆ isSO() [2/2]

◆ pseudoInverse()

◆ svd()