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   <command>
   {{Version|1}}
     <title>Numerical.EigenValuesAndAllVectors</title>
<command>
     <short_description>Eigenvalues and left and right eigenvectors of a matrix</short_description>
     <title>Num.EigenValuesAndAllVectors</title>
     <short_description>Computes eigenvalues and left and right eigenvectors of a matrix.</short_description>
<syntax>
<syntax>
$numerical.EigenValues(A:Matrix):List
Num.EigenValuesAndAllVectors(A:MAT):[B:MAT, C:MAT, D:MAT, E:MAT , F:MAT]
</syntax>
</syntax>
     <description>
     <description>
This function returns a List of three matrices, containing numerical approximation to A's eigenvalues and right and left eigenvectors.  
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
Therefore the input matrix A has to be rectangular!
<par/>
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!
This function returns a list of five matrices, containing numerical approximation of the eigenvalues of the matrix <tt>A</tt> and right and left eigenvectors.  
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.
<itemize>
<item>@param <em>A</em> A quadratic matrix with rational entries.</item>
<item>@return The output is a list of five matrices <tt>[B:MAT, C:MAT, D:MAT, E:MAT, F:MAT]</tt>. The first matrix <tt>B</tt> contains the complex eigenvalues of the matrix <tt>A</tt>, i.e. the first entry of a column is the real part and the second entry of the same column is the imaginary part of the eigenvalue. The matrices <tt>C</tt> and <tt>D</tt> represent the right eigenvectors of <tt>A</tt>, i.e. the <tt>j</tt>-th column of <tt>C</tt> contains the real part of the right eigenvector corresponding to eigenvalue <tt>j</tt> and the <tt>j</tt>-th column of D contains the imaginary part of the same right eigenvector corresponding to eigenvalue <tt>j</tt>. The matrices <tt>E</tt> and <tt>F</tt> store the left eigenvectors analogue to <tt>C</tt> and <tt>D</tt>.</item>
</itemize>
 
 
<example>
<example>
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);  
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);  
Numerical.EigenValuesAndAllVectors(A);
Dec(Num.EigenValuesAndAllVectors(A),3);
-- CoCoAServer: computing Cpu Time = 0.0031
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
-------------------------------
[Mat([
[Mat([
   [2038617447977453/70368744177664, 1593056728295919/4503599627370496, 0, 1717983664400761/562949953421312],
   ["28.970", "-13.677", "0.353", "0.353"],
   [-3850002255576293/281474976710656, 1593056728295919/4503599627370496, 0, -1717983664400761/562949953421312]
   ["0", "0", "3.051", "-3.051"]
]), Mat([
]), Mat([
   [-4846625556027553/9007199254740992, -675715895173401/1125899906842624, 6285574018989927/36028797018963968, -7024364631742823/18014398509481984],
   ["0.538", "-0.600", "0.389", "0.389"],
   [-5611119929071853/18014398509481984, -8025389267782659/36028797018963968, -630161806301403/4503599627370496, 7963794620848619/18014398509481984],
   ["0.311", "-0.222", "-0.442", "-0.442"],
   [-3851121972702563/9007199254740992, 6293666352540409/36028797018963968, -2394868378529203/9007199254740992, -1824257157284653/36028797018963968],
   ["0.427", "0.174", "0.050", "0.050"],
   [-5910799605047357/9007199254740992, 6738448111784605/9007199254740992, 6552680769135833/9007199254740992, 0]
   ["0.656", "0.748", "0", "0"]
]), Mat([
]), Mat([
   [-7110239176083849/18014398509481984, -5241040126502889/9007199254740992, -569232410323621/18014398509481984, 4695168387448581/18014398509481984],
   ["0", "0", "-0.174", "0.174"],
   [-7846388397589843/18014398509481984, -3981313256671163/9007199254740992, -2719422585742633/9007199254740992, -4930385173711605/9007199254740992],
  ["0", "0", "0.139", "-0.139"],
   [-3437594604471165/4503599627370496, 2800381393796867/4503599627370496, 6128985174171139/9007199254740992, 0],
  ["0", "0", "0.265", "-0.265"],
   [-1207381852306067/4503599627370496, 634514467740541/2251799813685248, -2469130937097749/9007199254740992, 6644460631770309/144115188075855872]
  ["0", "0", "-0.727", "0.727"]
]), Mat([
   ["0.394", "-0.581", "0.260", "0.260"],
  ["0.435", "-0.442", "-0.547", "-0.547"],
   ["0.763", "0.621", "0", "0"],
  ["0.268", "0.281", "0.046", "0.046"]
]), Mat([
  ["0", "0", "-0.031", "0.031"],
  ["0", "0", "-0.301", "0.301"],
   ["0", "0", "0.680", "-0.680"],
  ["0", "0", "-0.274", "0.274"]
])]
])]
-------------------------------
-------------------------------
Line 34: Line 50:
     </description>
     </description>
     <seealso>
     <seealso>
       <see>Introduction to CoCoAServer</see>
       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
       <see>Numerical.QR</see>
       <see>ApCoCoA-1:Num.QR|Num.QR</see>
       <see>Numerical.SVD</see>
       <see>ApCoCoA-1:Num.SVD|Num.SVD</see>
       <see>Numerical.EigenValues</see>
       <see>ApCoCoA-1:Num.EigenValues|Num.EigenValues</see>
       <see>Numerical.EigenValuesAndVectors</see>
       <see>ApCoCoA-1:Num.EigenValuesAndVectors|Num.EigenValuesAndVectors</see>
     </seealso>
     </seealso>
     <wiki-category>Package_Numerical</wiki-category>
    <types>
      <type>apcocoaserver</type>
      <type>matrix</type>
    </types>
    <key>EigenValuesAndAllVectors</key>
    <key>Num.EigenValuesAndAllVectors</key>
    <key>numerical.eigenvaluesandallvectors</key>
     <wiki-category>ApCoCoA-1:Package_numerical</wiki-category>
   </command>
   </command>

Latest revision as of 13:47, 29 October 2020

This article is about a function from ApCoCoA-1.

Num.EigenValuesAndAllVectors

Computes eigenvalues and left and right eigenvectors of a matrix.

Syntax

Num.EigenValuesAndAllVectors(A:MAT):[B:MAT, C:MAT, D:MAT, E:MAT , F:MAT]

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function returns a list of five matrices, containing numerical approximation of the eigenvalues of the matrix A and right and left eigenvectors.

  • @param A A quadratic matrix with rational entries.

  • @return The output is a list of five matrices [B:MAT, C:MAT, D:MAT, E:MAT, F:MAT]. The first matrix B contains the complex eigenvalues of the matrix A, i.e. the first entry of a column is the real part and the second entry of the same column is the imaginary part of the eigenvalue. The matrices C and D represent the right eigenvectors of A, i.e. the j-th column of C contains the real part of the right eigenvector corresponding to eigenvalue j and the j-th column of D contains the imaginary part of the same right eigenvector corresponding to eigenvalue j. The matrices E and F store the left eigenvectors analogue to C and D.


Example

A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); 
Dec(Num.EigenValuesAndAllVectors(A),3);
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
[Mat([
  ["28.970", "-13.677", "0.353", "0.353"],
  ["0", "0", "3.051", "-3.051"]
]), Mat([
  ["0.538", "-0.600", "0.389", "0.389"],
  ["0.311", "-0.222", "-0.442", "-0.442"],
  ["0.427", "0.174", "0.050", "0.050"],
  ["0.656", "0.748", "0", "0"]
]), Mat([
  ["0", "0", "-0.174", "0.174"],
  ["0", "0", "0.139", "-0.139"],
  ["0", "0", "0.265", "-0.265"],
  ["0", "0", "-0.727", "0.727"]
]), Mat([
  ["0.394", "-0.581", "0.260", "0.260"],
  ["0.435", "-0.442", "-0.547", "-0.547"],
  ["0.763", "0.621", "0", "0"],
  ["0.268", "0.281", "0.046", "0.046"]
]), Mat([
  ["0", "0", "-0.031", "0.031"],
  ["0", "0", "-0.301", "0.301"],
  ["0", "0", "0.680", "-0.680"],
  ["0", "0", "-0.274", "0.274"]
])]
-------------------------------

See also

Introduction to CoCoAServer

Num.QR

Num.SVD

Num.EigenValues

Num.EigenValuesAndVectors




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