Performs eigen decomposition of a matrix. More...

#include <SymEigen.h>

Public Member Functions
	SymEigen (int m)
template<int R, int C, typename B >
	SymEigen (const Matrix< R, C, Precision, B > &m)
template<int R, int C, typename B >
void	compute (const Matrix< R, C, Precision, B > &m)
template<int S, typename P , typename B >
Vector< Size, Precision >	backsub (const Vector< S, P, B > &rhs) const
template<int R, int C, typename P , typename B >
Matrix< Size, C, Precision >	backsub (const Matrix< R, C, P, B > &rhs) const
Matrix< Size, Size, Precision >	get_pinv (const double condition=Internal::symeigen_condition_no) const
Vector< Size, Precision >	get_inv_diag (const double condition) const
Matrix< Size, Size, Precision > &	get_evectors ()
const Matrix< Size, Size, Precision > &	get_evectors () const
Vector< Size, Precision > &	get_evalues ()
const Vector< Size, Precision > &	get_evalues () const
bool	is_posdef () const
bool	is_negdef () const
Precision	get_determinant () const
Matrix< Size, Size, Precision >	get_sqrtm () const
Matrix< Size, Size, Precision >	get_isqrtm (const double condition=Internal::symeigen_condition_no) const

Detailed Description

template<int Size = Dynamic, typename Precision = double>
class TooN::SymEigen< Size, Precision >

Performs eigen decomposition of a matrix.

Real symmetric (and hence square matrices) can be decomposed into

$M = U \times \Lambda \times U^T$

where $U$ is an orthogonal matrix (and hence $U^T = U^{-1}$ ) whose columns are the eigenvectors of $M$ and $\Lambda$ is a diagonal matrix whose entries are the eigenvalues of $M$ . These quantities are often of use directly, and can be obtained as follows:

// construct M
Matrix<3> M(3,3);
M[0]=makeVector(4,0,2);
M[1]=makeVector(0,5,3);
M[2]=makeVector(2,3,6);

// create the eigen decomposition of M
SymEigen<3> eigM(M);
cout << "A=" << M << endl;
cout << "(E,v)=eig(A)" << endl;
// print the smallest eigenvalue
cout << "v[0]=" << eigM.get_evalues()[0] << endl;
// print the associated eigenvector
cout << "E[0]=" << eigM.get_evectors()[0] << endl;

Further, provided the eigenvalues are nonnegative, the square root of a matrix and its inverse can also be obtained,

// print the square root of the matrix.
cout << "R=sqrtm(A)=" << eigM.get_sqrtm() << endl;
// print the square root of the matrix squared.
cout << "(should equal A), R^T*R="
     << eigM.get_sqrtm().T() * eigM.get_sqrtm() << endl;
// print the inverse of the matrix.
cout << "A^-1=" << eigM.get_pinv() << endl;
// print the inverse square root of the matrix.
cout << "C=isqrtm(A)=" << eigM.get_isqrtm() << endl;
// print the inverse square root of the matrix squared.
cout << "(should equal A^-1), C^T*C="
     << eigM.get_isqrtm().T() * eigM.get_isqrtm() << endl;

This decomposition is very similar to the SVD (q.v.), and can be used to solve equations using backsub() or get_pinv(), with the same treatment of condition numbers.

SymEigen<> (= SymEigen<-1>) can be used to create an eigen decomposition whose size is determined at run-time.

Constructor & Destructor Documentation

SymEigen ( int m )

Initialise this eigen decomposition but do no immediately perform a decomposition.

Parameters:

m	The size of the matrix to perform the eigen decomposition on.

SymEigen ( const Matrix< R, C, Precision, B > & m )

Construct the eigen decomposition of a matrix.

This initialises the class, and performs the decomposition immediately.

References SymEigen< Size, Precision >::compute().

Member Function Documentation

Vector<Size, Precision> backsub ( const Vector< S, P, B > & rhs ) const

Calculate result of multiplying the (pseudo-)inverse of M by a vector.

For a vector $b$ , this calculates $M^{\dagger}b$ by back substitution (i.e. without explictly calculating the (pseudo-)inverse). See the SVD detailed description for a description of condition variables.

References SymEigen< Size, Precision >::get_inv_diag(), and Matrix< Rows, Cols, Precision, Layout >::T().

Matrix<Size,C, Precision> backsub ( const Matrix< R, C, P, B > & rhs ) const

Calculate result of multiplying the (pseudo-)inverse of M by another matrix.

For a matrix $A$ , this calculates $M^{\dagger}A$ by back substitution (i.e. without explictly calculating the (pseudo-)inverse). See the SVD detailed description for a description of condition variables.

References SymEigen< Size, Precision >::get_inv_diag(), and Matrix< Rows, Cols, Precision, Layout >::T().

Matrix<Size, Size, Precision> get_pinv ( const double condition = Internal::symeigen_condition_no ) const

Calculate (pseudo-)inverse of the matrix.

This is not usually needed: if you need the inverse just to multiply it by a matrix or a vector, use one of the backsub() functions, which will be faster. See the SVD detailed description for a description of the pseudo-inverse and condition variables.

References SymEigen< Size, Precision >::get_inv_diag(), and Matrix< Rows, Cols, Precision, Layout >::T().

Vector<Size, Precision> get_inv_diag ( const double condition ) const

Calculates the reciprocals of the eigenvalues of the matrix.

The vector invdiag lists the eigenvalues in order, from the largest (i.e. smallest reciprocal) to the smallest. These are also the diagonal values of the matrix $Lambda^{-1}$ . Any eigenvalues which are too small are set to zero (see the SVD detailed description for a description of the and condition variables).

References Vector< Size, Precision, Base >::size().

Referenced by SymEigen< Size, Precision >::backsub(), and SymEigen< Size, Precision >::get_pinv().

Matrix<Size,Size,Precision>& get_evectors ( )

Returns the eigenvectors of the matrix.

This returns $U^T$ , so that the rows of the matrix are the eigenvectors, which can be extracted using usual Matrix::operator[]() subscript operator. They are returned in order of the size of the corresponding eigenvalue, i.e. the vector with the largest eigenvalue is first.

Vector<Size, Precision>& get_evalues ( )

Returns the eigenvalues of the matrix.

The eigenvalues are listed in order, from the smallest to the largest. These are also the diagonal values of the matrix $\Lambda$ .

Matrix<Size, Size, Precision> get_sqrtm ( ) const

Calculate the square root of a matrix which is a matrix M such that M.T*M=A.

References Vector< Size, Precision, Base >::size(), TooN::sqrt(), and Matrix< Rows, Cols, Precision, Layout >::T().

Matrix<Size, Size, Precision> get_isqrtm ( const double condition = Internal::symeigen_condition_no ) const

Calculate the inverse square root of a matrix which is a matrix M such that M.T*M=A^-1.

Any square-rooted eigenvalues which are too small are set to zero (see the SVD detailed description for a description of the condition variables).

References Vector< Size, Precision, Base >::size(), TooN::sqrt(), and Matrix< Rows, Cols, Precision, Layout >::T().

Public Member Functions

Detailed Description

template<int Size = Dynamic, typename Precision = double> class TooN::SymEigen< Size, Precision >

Constructor & Destructor Documentation

Member Function Documentation

template<int Size = Dynamic, typename Precision = double>
class TooN::SymEigen< Size, Precision >