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

## Num.EigenValuesAndVectors

Computes the eigenvalues and eigenvectors of a matrix.

### Syntax

```Num.EigenValuesAndVectors(A:Matrix):[B:Matrix, C:Matrix, 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 numerical approximation to the eigenvalues of A and (right hand) eigenvectors.

• @param A A square matrix with rational entries.

• @return The output [B:Matrix, C:Matrix, D:Matrix] contains a matrix B, where each column contains one of A's eigenvalues. The first row contains the eigenvalue's real part, the second row the imaginary. The matrices C and D both have the same dimensions as A. Column j of matrix C contains the real part of the eigenvector corresponding to eigenvalue j and column j of matrix D contains the imaginary part of the eigenvector correspsonding to eigenvalue j.

To compute only the left hand eigenvectors apply this method 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([
[<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>0.394</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>]
])]
--------------------------------------------------------------
```