# 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([
[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]
])]
-------------------------------
```