# ApCoCoA-1:Num.EigenValuesAndAllVectors

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