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

## 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"]
])]
-------------------------------
```