# ApCoCoA: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([
[<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>]
])]
-------------------------------
```