Difference between revisions of "ApCoCoA-1:Num.EigenValuesAndAllVectors"

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<item>@param <em>A</em> A quadratic matrix with rational entries.</item>
 
<item>@param <em>A</em> A quadratic matrix with rational entries.</item>
<item>@return The output <tt>[B:MAT, C:MAT, D:MAT, E:MAT, F:MAT]</tt> contains a matrix <tt>B</tt>, where each column contains one complex eigenvalue of the matrix <tt>A</tt>. The first entry of a column contains the real part of complex eigenvalue, the second entry the imaginary part. The matrices <tt>C</tt>, <tt>D</tt>, <tt>E</tt> and <tt>F</tt> all have the same dimensions as <tt>A</tt>. Column j of matrix C contains the real part of the right eigenvector corresponding to eigenvalue j and column j of matrix D contains the imaginary part of the right eigenvector corresponding to eigenvalue j. The matrices E and F store the left eigenvectors in the same way as C and D.</item>
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<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>
 
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Revision as of 07:48, 8 July 2009

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([
  [<quotes>28.970</quotes>, <quotes>-13.677</quotes>, <quotes>0.353</quotes>, <quotes>0.353</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>3.051</quotes>, <quotes>-3.051</quotes>]
]), Mat([
  [<quotes>0.538</quotes>, <quotes>-0.600</quotes>, <quotes>0.389</quotes>, <quotes>0.389</quotes>],
  [<quotes>0.311</quotes>, <quotes>-0.222</quotes>, <quotes>-0.442</quotes>, <quotes>-0.442</quotes>],
  [<quotes>0.427</quotes>, <quotes>0.174</quotes>, <quotes>0.050</quotes>, <quotes>0.050</quotes>],
  [<quotes>0.656</quotes>, <quotes>0.748</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]
]), Mat([
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.174</quotes>, <quotes>0.174</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.139</quotes>, <quotes>-0.139</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.265</quotes>, <quotes>-0.265</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.727</quotes>, <quotes>0.727</quotes>]
]), Mat([
  [<quotes>0.394</quotes>, <quotes>-0.581</quotes>, <quotes>0.260</quotes>, <quotes>0.260</quotes>],
  [<quotes>0.435</quotes>, <quotes>-0.442</quotes>, <quotes>-0.547</quotes>, <quotes>-0.547</quotes>],
  [<quotes>0.763</quotes>, <quotes>0.621</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],
  [<quotes>0.268</quotes>, <quotes>0.281</quotes>, <quotes>0.046</quotes>, <quotes>0.046</quotes>]
]), Mat([
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.031</quotes>, <quotes>0.031</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.301</quotes>, <quotes>0.301</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.680</quotes>, <quotes>-0.680</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.274</quotes>, <quotes>0.274</quotes>]
])]
-------------------------------

See also

Introduction to CoCoAServer

Num.QR

Num.SVD

Num.EigenValues

Num.EigenValuesAndVectors