Difference between revisions of "ApCoCoA-1:Num.EigenValuesAndAllVectors"
From ApCoCoAWiki
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| − | <command> | + | {{Version|1}} |
| + | <command> | ||
<title>Num.EigenValuesAndAllVectors</title> | <title>Num.EigenValuesAndAllVectors</title> | ||
| − | <short_description>Computes eigenvalues and left and right eigenvectors of a matrix</short_description> | + | <short_description>Computes eigenvalues and left and right eigenvectors of a matrix.</short_description> |
<syntax> | <syntax> | ||
Num.EigenValuesAndAllVectors(A:MAT):[B:MAT, C:MAT, D:MAT, E:MAT , F:MAT] | Num.EigenValuesAndAllVectors(A:MAT):[B:MAT, C:MAT, D:MAT, E:MAT , F:MAT] | ||
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<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. | <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. | ||
<par/> | <par/> | ||
| − | This function returns a list of five matrices, containing numerical approximation | + | 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. |
<itemize> | <itemize> | ||
| − | <item>@param <em>A</em> A | + | <item>@param <em>A</em> A quadratic matrix with rational entries.</item> |
| − | <item>@return The output [B: | + | <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> | </itemize> | ||
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------------------------------- | ------------------------------- | ||
[Mat([ | [Mat([ | ||
| − | [ | + | ["28.970", "-13.677", "0.353", "0.353"], |
| − | [ | + | ["0", "0", "3.051", "-3.051"] |
]), Mat([ | ]), 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([ | ]), 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([ | ]), 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([ | ]), 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"] |
])] | ])] | ||
------------------------------- | ------------------------------- | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
| − | <see>Num.QR</see> | + | <see>ApCoCoA-1:Num.QR|Num.QR</see> |
| − | <see>Num.SVD</see> | + | <see>ApCoCoA-1:Num.SVD|Num.SVD</see> |
| − | <see>Num.EigenValues</see> | + | <see>ApCoCoA-1:Num.EigenValues|Num.EigenValues</see> |
| − | <see>Num.EigenValuesAndVectors</see> | + | <see>ApCoCoA-1:Num.EigenValuesAndVectors|Num.EigenValuesAndVectors</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
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<key>Num.EigenValuesAndAllVectors</key> | <key>Num.EigenValuesAndAllVectors</key> | ||
<key>numerical.eigenvaluesandallvectors</key> | <key>numerical.eigenvaluesandallvectors</key> | ||
| − | <wiki-category>Package_numerical</wiki-category> | + | <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
