ApCoCoA-1:Num.EigenValuesAndAllVectors

Numerical.EigenValuesAndAllVectors

eigenvalues and left and right eigenvectors of a matrix

Syntax

$numerical.EigenValues(A:Matrix):List

Description

This function returns a List of three matrices, containing numerical approximation to A's eigenvalues and right and left eigenvectors.

Therefore the input matrix A has to be rectangular!

It is implemented in the ApCoCoA server, so you need a running server. It was not implemented in version 0.99.4 or previous. Also please keep in mind this method is based on blas/Lapack's eigenvalue solver and uses floating point arithmetic. This is not an exact, algebraic method! The output list contains first a matrix B. Each row of B describe one of A's eigenvalues. The first column contains the eigenvalue's real part, the second the imaginary. The second element of the list is a matrix of the size of A, containing A's left hand eigenvectors, while the third element in the list is a matrix containing the right hand eigenvectors.

Example

A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); 
Numerical.EigenValuesAndAllVectors(A);
-- CoCoAServer: computing Cpu Time = 0.0031
-------------------------------
[Mat([
  [2038617447977453/70368744177664, 1593056728295919/4503599627370496, 0, 1717983664400761/562949953421312],
  [-3850002255576293/281474976710656, 1593056728295919/4503599627370496, 0, -1717983664400761/562949953421312]
]), Mat([
  [-4846625556027553/9007199254740992, -675715895173401/1125899906842624, 6285574018989927/36028797018963968, -7024364631742823/18014398509481984],
  [-5611119929071853/18014398509481984, -8025389267782659/36028797018963968, -630161806301403/4503599627370496, 7963794620848619/18014398509481984],
  [-3851121972702563/9007199254740992, 6293666352540409/36028797018963968, -2394868378529203/9007199254740992, -1824257157284653/36028797018963968],
  [-5910799605047357/9007199254740992, 6738448111784605/9007199254740992, 6552680769135833/9007199254740992, 0]
]), Mat([
  [-7110239176083849/18014398509481984, -5241040126502889/9007199254740992, -569232410323621/18014398509481984, 4695168387448581/18014398509481984],
  [-7846388397589843/18014398509481984, -3981313256671163/9007199254740992, -2719422585742633/9007199254740992, -4930385173711605/9007199254740992],
  [-3437594604471165/4503599627370496, 2800381393796867/4503599627370496, 6128985174171139/9007199254740992, 0],
  [-1207381852306067/4503599627370496, 634514467740541/2251799813685248, -2469130937097749/9007199254740992, 6644460631770309/144115188075855872]
])]
-------------------------------

See also

Introduction to CoCoAServer

Numerical.QR

Numerical.SVD

Numerical.EigenValues

Numerical.EigenValuesAndVectors