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

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.factorization.eigen.CharacteristicPolynomial

public class CharacteristicPolynomial
extends java.lang.Object

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

Constructor Summary

CharacteristicPolynomial(Matrix A)
Construct a CharacteristicPolynomial from a square matrix.

Method Summary

Polynomial	characteristicPolynomial() Get a copy of the characteristic polynomial.
NumberList	eigenvalues() 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 a CharacteristicPolynomial from a square matrix.

Parameters:

A - a matrix

Throws:

java.lang.IllegalArgumentException - if A is not square

Method Detail

characteristicPolynomial

public Polynomial characteristicPolynomial()

Get a copy of the characteristic polynomial.

Returns:

a copy of the characteristic polynomial

eigenvalues

public NumberList eigenvalues()

Get all the eigenvalues.

Returns:

the eigenvalues

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object