smile.math.matrix

Class EigenValueDecomposition

java.lang.Object

smile.math.matrix.EigenValueDecomposition

public class EigenValueDecomposition
extends 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.

Author:

Haifeng Li

Method Summary

Modifier and Type	Method and Description
static EigenValueDecomposition	decompose(double[][] A) Full eigen value decomposition of a square matrix.
static EigenValueDecomposition	decompose(double[][] A, boolean symmetric) Full eigen value decomposition of a square matrix.
static EigenValueDecomposition	decompose(double[][] A, boolean symmetric, boolean onlyValues) Full eigen value decomposition of a square matrix.
static EigenValueDecomposition	decompose(IMatrix A, int k) Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
static EigenValueDecomposition	decompose(IMatrix A, int k, double kappa) Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
static double	eigen(IMatrix A, double[] v) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(IMatrix A, double[] v, double tol) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(IMatrix A, double[] v, double p, double tol) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(IMatrix A, double[] v, double p, double tol, int maxIter) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(IMatrix A, double[] v, double tol, int maxIter) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
double[][]	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.
double[][]	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.
static double[]	pagerank(IMatrix A) Calculate the page rank vector.
static double[]	pagerank(IMatrix A, double[] v) Calculate the page rank vector.
static double[]	pagerank(IMatrix A, double[] v, double damping, double tol, int maxIter) Calculate the page rank vector.

Methods inherited from class java.lang.Object

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

Method Detail

getEigenVectors

public double[][] 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 double[][] 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.

eigen

public static double eigen(IMatrix A,
                           double[] v)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

Returns:

the largest eigen value.

eigen

public static double eigen(IMatrix A,
                           double[] v,
                           double tol)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

tol - the desired convergence tolerance.

Returns:

the largest eigen value.

eigen

public static double eigen(IMatrix A,
                           double[] v,
                           double tol,
                           int maxIter)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

tol - the desired convergence tolerance.

maxIter - the maximum number of iterations in case that the algorithm does not converge.

Returns:

the largest eigen value.

eigen

public static double eigen(IMatrix A,
                           double[] v,
                           double p,
                           double tol)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

p - the origin in the shifting power method. A - pI will be used in the iteration to accelerate the method. p should be such that |(λ₂ - p) / (λ₁ - p)| < |λ₂ / λ₁|, where λ₂ is the second largest eigenvalue in magnitude. If we known the eigenvalue spectrum of A, (λ₂ + λ_n)/2 is the optimal choice of p, where λ_n is the smallest eigenvalue in magnitude. Good estimates of λ₂ are more difficult to compute. However, if μ is an approximation to largest eigenvector, then using any x₀ such that x₀*μ = 0 as the initial vector for a few iterations may yield a reasonable estimate of λ₂.

tol - the desired convergence tolerance.

Returns:

the largest eigen value.

eigen

public static double eigen(IMatrix A,
                           double[] v,
                           double p,
                           double tol,
                           int maxIter)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

p - the origin in the shifting power method. A - pI will be used in the iteration to accelerate the method. p should be such that |(λ₂ - p) / (λ₁ - p)| < |λ₂ / λ₁|, where λ₂ is the second largest eigenvalue in magnitude. If we known the eigenvalue spectrum of A, (λ₂ + λ_n)/2 is the optimal choice of p, where λ_n is the smallest eigenvalue in magnitude. Good estimates of λ₂ are more difficult to compute. However, if μ is an approximation to largest eigenvector, then using any x₀ such that x₀*μ = 0 as the initial vector for a few iterations may yield a reasonable estimate of λ₂.

tol - the desired convergence tolerance.

maxIter - the maximum number of iterations in case that the algorithm does not converge.

Returns:

the largest eigen value.

pagerank

public static double[] pagerank(IMatrix A)

Calculate the page rank vector.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

Returns:

the page rank vector.

pagerank

public static double[] pagerank(IMatrix A,
                                double[] v)

Calculate the page rank vector.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - the teleportation vector.

Returns:

the page rank vector.

pagerank

public static double[] pagerank(IMatrix A,
                                double[] v,
                                double damping,
                                double tol,
                                int maxIter)

Calculate the page rank vector.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - the teleportation vector.

damping - the damper factor.

tol - the desired convergence tolerance.

maxIter - the maximum number of iterations in case that the algorithm does not converge.

Returns:

the page rank vector.

decompose

public static EigenValueDecomposition decompose(IMatrix A,
                                                int k)

Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.

decompose

public static EigenValueDecomposition decompose(IMatrix A,
                                                int k,
                                                double kappa)

Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.

kappa - relative accuracy of ritz values acceptable as eigenvalues.

decompose

public static EigenValueDecomposition decompose(double[][] A)

Full eigen value decomposition of a square matrix. Note that the input matrix will be altered during decomposition.

Parameters:

A - square matrix which will be altered during decomposition.

decompose

public static EigenValueDecomposition decompose(double[][] A,
                                                boolean symmetric)

Full eigen value decomposition of a square matrix. Note that the input matrix will be altered during decomposition.

Parameters:

A - square matrix which will be altered during decomposition.

symmetric - if true, the matrix is assumed to be symmetric.

decompose

public static EigenValueDecomposition decompose(double[][] A,
                                                boolean symmetric,
                                                boolean onlyValues)

Full eigen value decomposition of a square matrix. Note that the input matrix will be altered during decomposition.

Parameters:

A - square matrix which will be altered during decomposition.

symmetric - if true, the matrix is assumed to be symmetric.

onlyValues - if true, only compute eigenvalues; the default is to compute eigenvectors also.