borgmater - 2 months ago 16

Python Question

As a current task, I need to calculate eigenvalues and eigenvectors for a 120*120 matrix. For start, I tested those calculations on a simple 2 by 2 matrix in both Java (Apache Commons Math library) and Python 2.7 (Numpy library). I have a problem with eigenvector values not matching, as show below :

`//Java`

import org.apache.commons.math3.linear.EigenDecomposition;

import org.apache.commons.math3.linear.MatrixUtils;

import org.apache.commons.math3.linear.RealMatrix;

public class TemporaryTest {

public static void main(String[] args) {

double[][] testArray = {{2, -1}, {1, 1}};

RealMatrix testMatrix = MatrixUtils.createRealMatrix(testArray);

EigenDecomposition decomposition = new EigenDecomposition (testMatrix);

System.out.println("eigenvector[0] = " + decomposition.getEigenvector(0));

System.out.println("eigenvector[1] = " + decomposition.getEigenvector(1));

}

Output of eigenvectors are shown as {real_value + imaginary_value; real_value + imaginary_value}:

`//Java output`

eigenvector[0] = {-0.8660254038; 0}

eigenvector[1] = {0.5; 1}

Same code in Python, but using Numpy library:

`# Python`

import numpy as np

from numpy import linalg as LA

w, v = LA.eig(np.array([[2, -1], [1, 1]]))

print (v[:, 0])

print (v[:, 1])

Output of eigenvectors in Python are shown similarly, [real+imag real+imag]:

`[ 0.35355339+0.61237244j 0.70710678+0.j ]`

[ 0.35355339-0.61237244j 0.70710678-0.j ]

My concern is, why are those vectors different ? Is there something that I am missing ? Ty for any kind of help or advice

Answer

In Apache Commons Math 3, `EigenDecomposition`

accepts nonsymmetric matrices, but it returns results using the classes `RealVector`

and `RealMatrix`

. To get the actual complex results, you have to combine the appropriate real results into complex conjugate pairs.

In the case of the eigenvectors, you got:

```
eigenvector[0] = {-0.8660254038; 0}
eigenvector[1] = {0.5; 1}
```

Both those vectors are associated with the complex conjugate pair of eigenvalues `getRealEigenvalue(0) + getImagEigenvalue(0)*i`

and `getRealEigenvalue(1) + getImagEigenvalue(1)*i`

, but those vectors are not the actual eigenvectors. The actual eigenvectors are the complex conjugate pairs
`eigenvector[0] + eigenvector[1]*i`

and `eigenvector[0] - eigenvector[1]*i`

.

Those vectors still don't "match" the results returned by numpy, but that is because the two libraries have not used the same normalization. Eigenvectors are not unique; an eigenvector multiplied by any nonzero scalar (including a complex scalar) is still an eigenvector. The only difference between the Java result and the numpy result is a scalar multiplier.

For convenience, I'll convert the floating point values to their exact values. That is, `-0.8660254038`

is the floating point approximation of `-sqrt(3)/2`

. The Java math library is giving the following eigenvectors:

```
[-sqrt(3)/2 + (1/2)*i] and [-sqrt(3)/2 - (1/2)*i]
[ i ] [ -i ]
```

If you multiply the first eigenvector by -(sqrt(2)/2)*i and the second by (sqrt(2)/2)*i, you'll get the eigenvectors that were return by numpy.

Here's an ipython session with that calculation. `v1`

and `v2`

are the vectors shown above.

```
In [20]: v1 = np.array([-np.sqrt(3)/2 + 0.5j, 1j])
In [21]: v1
Out[21]: array([-0.8660254+0.5j, 0.0000000+1.j ])
In [22]: v2 = np.array([-np.sqrt(3)/2 - 0.5j, -1j])
In [23]: v2
Out[23]: array([-0.8660254-0.5j, 0.0000000-1.j ])
```

Multiply `v1`

by -(sqrt(2)/2)*i to get the first eigenvector returned by `numpy.linalg.eig`

:

```
In [24]: v1*(-np.sqrt(2)/2*1j)
Out[24]: array([ 0.35355339+0.61237244j, 0.70710678-0.j ])
```

Multiply `v2`

by (sqrt(2)/2)*i to get the second eigenvector returned by `numpy.linalg.eig`

:

```
In [25]: v2*(np.sqrt(2)/2*1j)
Out[25]: array([ 0.35355339-0.61237244j, 0.70710678+0.j ])
```

For convenience, here's a repeat of the numpy calculation. The columns of `evecs`

are the eigenvectors.

```
In [28]: evals, evecs = np.linalg.eig(a)
In [29]: evecs
Out[29]:
array([[ 0.35355339+0.61237244j, 0.35355339-0.61237244j],
[ 0.70710678+0.j , 0.70710678-0.j ]])
```