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 contains first of a matrix B, where the number of rows contains 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