public class EigenDecompositionextends ObjectCalculates the eigen decomposition of a real matrix.
The eigen decomposition of matrix A is a set of two matrices: V and D such that A = V × D × VT. A, V and D are all m × m matrices.
This class is similar in spirit to the
EigenvalueDecompositionclass from the JAMA library, with the following changes:
getVtmethod has been added,
getImagEigenvaluemethods to pick up a single eigenvalue have been added,
getEigenvectormethod to pick up a single eigenvector has been added,
getDeterminantmethod has been added.
getSolvermethod has been added.
As of 3.1, this class supports general real matrices (both symmetric and non-symmetric):
If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal, i.e. A = V.multiply(D.multiply(V.transpose())) and V.multiply(V.transpose()) equals the identity matrix.
If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks:
[lambda, mu ] [ -mu, lambda]The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.multiply(V) equals V.multiply(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon the condition of V.
This implementation is based on the paper by A. Drubrulle, R.S. Martin and J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971) Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag, New-York