Package org.apache.commons.math4.legacy.linear

Class EigenDecomposition

  • public class EigenDecomposition
extends Object
    Calculates the eigen decomposition of a real matrix.

    The eigen decomposition of matrix A is a set of two matrices: V and D such that A = V × D × VT. A, V and D are all m × m matrices.

    This class is similar in spirit to the EigenvalueDecomposition class from the JAMA library, with the following changes:

    As of 3.1, this class supports general real matrices (both symmetric and non-symmetric):

    If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal, i.e. A = V.multiply(D.multiply(V.transpose())) and V.multiply(V.transpose()) equals the identity matrix.

    If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks: 

        [lambda, mu    ]
    [   -mu, lambda]
    The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.multiply(V) equals V.multiply(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon the condition of V.

    This implementation is based on the paper by A. Drubrulle, R.S. Martin and J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971) Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag, New-York.

    Since:
    2.0 (changed to concrete class in 3.0)
    See Also:
    MathWorld, Wikipedia

    • Constructor Detail

      • EigenDecomposition

        public EigenDecomposition​(double[] main,
                          double[] secondary)
        Calculates the eigen decomposition of the symmetric tridiagonal matrix. The Householder matrix is assumed to be the identity matrix.
        Parameters:
        main - Main diagonal of the symmetric tridiagonal form.
        secondary - Secondary of the tridiagonal form.
        Throws:
        MaxCountExceededException - if the algorithm fails to converge.
        Since:
        3.1

    • Method Detail

      • getV

        public RealMatrix getV()
        Gets the matrix V of the decomposition. V is an orthogonal matrix, i.e. its transpose is also its inverse. The columns of V are the eigenvectors of the original matrix. No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).
        Returns:
        the V matrix.

      • getD

        public RealMatrix getD()
        Gets the block diagonal matrix D of the decomposition. D is a block diagonal matrix. Real eigenvalues are on the diagonal while complex values are on 2x2 blocks { {real +imaginary}, {-imaginary, real} }.
        Returns:
        the D matrix.
        See Also:
        getRealEigenvalues(), getImagEigenvalues()

      • getVT

        public RealMatrix getVT()
        Gets the transpose of the matrix V of the decomposition. V is an orthogonal matrix, i.e. its transpose is also its inverse. The columns of V are the eigenvectors of the original matrix. No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).
        Returns:
        the transpose of the V matrix.

      • hasComplexEigenvalues

        public boolean hasComplexEigenvalues()
        Returns whether the calculated eigen values are complex or real.

        The method performs a zero check for each element of the getImagEigenvalues() array and returns true if any element is not equal to zero.

        Returns:
        true if the eigen values are complex, false otherwise
        Since:
        3.1

      • getImagEigenvalue

        public double getImagEigenvalue​(int i)
        Gets the imaginary part of the ith eigenvalue of the original matrix.
        Parameters:
        i - Index of the eigenvalue (counting from 0).
        Returns:
        the imaginary part of the ith eigenvalue of the original matrix.
        See Also:
        getD(), getImagEigenvalues(), getRealEigenvalue(int)

      • getEigenvector

        public RealVector getEigenvector​(int i)
        Gets a copy of the ith eigenvector of the original matrix.
        Parameters:
        i - Index of the eigenvector (counting from 0).
        Returns:
        a copy of the ith eigenvector of the original matrix.
        See Also:
        getD()

      • getDeterminant

        public double getDeterminant()
        Computes the determinant of the matrix.
        Returns:
        the determinant of the matrix.

      • getSquareRoot

        public RealMatrix getSquareRoot()
        Computes the square-root of the matrix. This implementation assumes that the matrix is symmetric and positive definite.
        Returns:
        the square-root of the matrix.
        Throws:
        MathUnsupportedOperationException - if the matrix is not symmetric or not positive definite.
        Since:
        3.1