smile.math.matrix

Class EVD

java.lang.Object

smile.math.matrix.EVD

public class EVD
extends java.lang.Object

Eigen decomposition of a real matrix. Eigen decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors:

A = V*D*V^-1

If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal.

Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation

A x = λ x

for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x.

The word eigenvector formally refers to the right eigenvector, which is defined by the above eigenvalue equation A x = λ x, and is the most commonly used eigenvector. However, the left eigenvector exists as well, and is defined by x A = λ x.

Let A be a real n-by-n matrix with strictly positive entries a_ij > 0. Then the following statements hold.

1. There is a positive real number r, called the Perron-Frobenius eigenvalue, such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) is strictly smaller than r in absolute value, |λ| < r.
2. The Perron-Frobenius eigenvalue is simple: r is a simple root of the characteristic polynomial of A. Consequently, both the right and the left eigenspace associated to r is one-dimensional.
3. There exists a left eigenvector v of A associated with r (row vector) having strictly positive components. Likewise, there exists a right eigenvector w associated with r (column vector) having strictly positive components.
4. The left eigenvector v (respectively right w) associated with r, is the only eigenvector which has positive components, i.e. for all other eigenvectors of A there exists a component which is not positive.

A stochastic matrix, probability matrix, or transition matrix is used to describe the transitions of a Markov chain. A right stochastic matrix is a square matrix each of whose rows consists of nonnegative real numbers, with each row summing to 1. A left stochastic matrix is a square matrix whose columns consist of nonnegative real numbers whose sum is 1. A doubly stochastic matrix where all entries are nonnegative and all rows and all columns sum to 1. A stationary probability vector π is defined as a vector that does not change under application of the transition matrix; that is, it is defined as a left eigenvector of the probability matrix, associated with eigenvalue 1: πP = π. The Perron-Frobenius theorem ensures that such a vector exists, and that the largest eigenvalue associated with a stochastic matrix is always 1. For a matrix with strictly positive entries, this vector is unique. In general, however, there may be several such vectors.

Constructor Summary

Constructors

Constructor and Description
EVD(DenseMatrix V, double[] d) Private constructor.
EVD(DenseMatrix V, double[] d, double[] e) Private constructor.

Method Summary

Modifier and Type	Method and Description
DenseMatrix	getD() Returns the block diagonal eigenvalue matrix whose diagonal are the real part of eigenvalues, lower subdiagonal are positive imaginary parts, and upper subdiagonal are negative imaginary parts.
double[]	getEigenValues() Returns the eigenvalues, ordered from largest to smallest.
DenseMatrix	getEigenVectors() Returns the eigenvector matrix, ordered by eigen values from largest to smallest.
double[]	getImagEigenValues() Returns the imaginary parts of the eigenvalues, ordered in real part from largest to smallest.
double[]	getRealEigenValues() Returns the real parts of the eigenvalues, ordered in real part from largest to smallest.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

EVD

public EVD(DenseMatrix V,
           double[] d)

Private constructor.

Parameters:

V - eigenvectors.

d - eigenvalues.

EVD

public EVD(DenseMatrix V,
           double[] d,
           double[] e)

Private constructor.

Parameters:

V - eigenvectors.

d - real part of eigenvalues.

e - imaginary part of eigenvalues.

Method Detail

getEigenVectors

public DenseMatrix getEigenVectors()

Returns the eigenvector matrix, ordered by eigen values from largest to smallest.

getEigenValues

public double[] getEigenValues()

Returns the eigenvalues, ordered from largest to smallest.

getRealEigenValues

public double[] getRealEigenValues()

Returns the real parts of the eigenvalues, ordered in real part from largest to smallest.

getImagEigenValues

public double[] getImagEigenValues()

Returns the imaginary parts of the eigenvalues, ordered in real part from largest to smallest.

getD

public DenseMatrix getD()

Returns the block diagonal eigenvalue matrix whose diagonal are the real part of eigenvalues, lower subdiagonal are positive imaginary parts, and upper subdiagonal are negative imaginary parts.