# ApCoCoA-1:Num.EigenValuesAndVectors

## Num.EigenValuesAndVectors

Computes the eigenvalues and eigenvectors of a matrix.

### Syntax

```Num.EigenValuesAndVectors(A:MAT):[B:MAT, C:MAT, D:Matrix]
```

### 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 three matrices containing the numerical approximations of the complex eigenvalues and right eigenvectors of A.

• @param A A quadratic matrix with rational entries.

• @return The output is a list of three matrices [B:Matrix, C:Matrix, D:Matrix]. 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.

In order to compute the left hand eigenvectors of A, apply this command to the transposed matrix of A (see Transposed).

#### Example

```A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);
Dec(Num.EigenValuesAndVectors(A),3);

-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
[Mat([
["28.970", "-13.677", "0.353", "0.353"],
["0", "0", "3.051", "-3.051"]
]), Mat([
["0.394", "-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"]
])]
--------------------------------------------------------------
```