ApCoCoA:Num.EigenValuesAndVectors

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 <command>
   <title>Num.EigenValuesAndVectors</title>
   <short_description>Computes the eigenvalues and eigenvectors of a matrix.</short_description>

<syntax> Num.EigenValuesAndVectors(A:MAT):[B:MAT, C:MAT, D:Matrix] </syntax>

   <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. <par/> This function returns a list of three matrices containing the numerical approximations of the complex eigenvalues and right eigenvectors of A.

<itemize> <item>@param A A quadratic matrix with rational entries.</item> <item>@return The output is a list of three matrices [B:Matrix, C:Matrix, D:Matrix]. 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.</item> </itemize>

In order to compute the left hand eigenvectors of A, apply this command to the transposed matrix of A (see <ref>Transposed</ref>).

<example> A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); Dec(Num.EigenValuesAndVectors(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>0.394</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>]

])]


</example>

   </description>
   <seealso>
     <see>Introduction to CoCoAServer</see>
     <see>Num.QR</see>
     <see>Num.SVD</see>
     <see>Num.EigenValues</see>
     <see>Num.EigenValuesAndAllVectors</see>
   </seealso>
   <types>
     <type>apcocoaserver</type>
     <type>matrix</type>
   </types>
   <key>Num.EigenValuesAndVectors</key>
   <key>EigenValuesAndVectors</key>
   <key>numerical.eigenvaluesandvectors</key>
   <wiki-category>Package_numerical</wiki-category>
 </command>