com.numericalmethod.suanshu.matrix.doubles.factorization.eigen

Class CharacteristicPolynomial

java.lang.Object

com.numericalmethod.suanshu.matrix.doubles.factorization.eigen.CharacteristicPolynomial

All Implemented Interfaces:

Spectrum, java.io.Serializable

public class CharacteristicPolynomial
extends java.lang.Object
implements Spectrum

The characteristic polynomial of a square matrix is the function

 p(λ) = det(A - λI)
 

The zeros of this polynomial are the eigenvalues of A. That is, λ being an eigenvalue of A is equivalent to stating that the system of linear equations

 (A - λI) v = 0
 

where I is an identity matrix, has a non-zero solution v (namely an eigenvector).

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial, that is, p(A) = 0 .

See Also:

• Wikipedia: Eigenvalue algorithm
• Wikipedia: Cayley–Hamilton theorem

, Serialized Form

Constructor Summary

Constructors

Constructor and Description
CharacteristicPolynomial(Matrix A) Construct the characteristic polynomial for a square matrix.

Method Summary

Modifier and Type	Method and Description
Polynomial	getCharacteristicPolynomial() Get the characteristic polynomial.
java.util.List<java.lang.Number>	getEigenvalues() Get all the eigenvalues.
java.lang.String	toString()

Methods inherited from class java.lang.Object

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

Constructor Detail

CharacteristicPolynomial

public CharacteristicPolynomial(Matrix A)

Construct the characteristic polynomial for a square matrix.

Parameters:

A - a square matrix

Throws:

java.lang.IllegalArgumentException - if A is not square

Method Detail

getCharacteristicPolynomial

public Polynomial getCharacteristicPolynomial()

Get the characteristic polynomial.

Returns:

the characteristic polynomial

getEigenvalues

public java.util.List<java.lang.Number> getEigenvalues()

Description copied from interface: Spectrum

Get all the eigenvalues.

Specified by:

getEigenvalues in interface Spectrum

Returns:

the eigenvalues

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object