smile.math.matrix

Class FloatMatrix.EVD

java.lang.Object

smile.math.matrix.FloatMatrix.EVD

All Implemented Interfaces:

java.io.Serializable

Enclosing class:

FloatMatrix

public static class FloatMatrix.EVD
extends java.lang.Object
implements java.io.Serializable

Eigenvalue decomposition. 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.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
FloatMatrix	Vl The left eigenvectors.
FloatMatrix	Vr The right eigenvectors.
float[]	wi The imaginary part of eigenvalues.
float[]	wr The real part of eigenvalues.

Constructor Summary

Constructors

Constructor and Description
EVD(float[] wr, float[] wi, FloatMatrix Vl, FloatMatrix Vr) Constructor.
EVD(float[] w, FloatMatrix V) Constructor.

Method Summary

Modifier and Type	Method and Description
FloatMatrix	diag() 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.
FloatMatrix.EVD	sort() Sorts the eigenvalues in descending order and reorders the corresponding eigenvectors.

Methods inherited from class java.lang.Object

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

Field Detail

wr

public final float[] wr

The real part of eigenvalues. By default the eigenvalues and eigenvectors are not always in sorted order. The sort function puts the eigenvalues in descending order and reorder the corresponding eigenvectors.

wi

public final float[] wi

The imaginary part of eigenvalues.

Vl

public final FloatMatrix Vl

The left eigenvectors.

Vr

public final FloatMatrix Vr

The right eigenvectors.

Constructor Detail

EVD

public EVD(float[] w,
           FloatMatrix V)

Constructor.

Parameters:

w - eigenvalues.

V - eigenvectors.

EVD

public EVD(float[] wr,
           float[] wi,
           FloatMatrix Vl,
           FloatMatrix Vr)

Constructor.

Parameters:

wr - the real part of eigenvalues.

wi - the imaginary part of eigenvalues.

Vl - the left eigenvectors.

Vr - the right eigenvectors.

Method Detail

diag

public FloatMatrix diag()

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.

sort

public FloatMatrix.EVD sort()

Sorts the eigenvalues in descending order and reorders the corresponding eigenvectors.