realEigenvalues

lollielgelParticipant 2010-12-16 at 9:52 am
How does Eigen deal with multiple roots or complex eigenValues? Some examples would be great.

I’m trying to do something like this:
```
 
 DenseMatrix sigma = new DenseMatrix(new double[][]{ 
 {1., 0.}, 
 {0., 1.}}); 
Eigen sigmaEigen = new Eigen(sigma); 
double[] sigmaEigenvalues = sigmaEigen.realEigenvalues(); 
```
and would like sigmaEigenvalues to contain 2 identical real numbers {1, 1}.

Thanks.

Haksun LiModerator 2010-12-16 at 10:03 am

The property of an eigenvalue is returned in the structure

Eigen.Property

E.g.,

<br />
       Matrix A = new DenseMatrix(new double[][]{<br />
                   {1, 0},<br />
                   {0, 1}<br />
               });<br />
       Eigen eigen = new Eigen(A);<br />
<br />
       assertTrue(eigen.eigenvalues().hasApproximately(1, 1));<br />
<br />
       assertEquals(1, eigen.size());<br />
       assertEquals(1, eigen.realEigenvalues().length);<br />
<br />
       for (int i = 0; i < eigen.size(); ++i) {<br />
           double eigenvalue = eigen.realEigenvalues()<i>;<br />
           Property property = eigen.property(eigenvalue);<br />
           assertEquals(eigenvalue, property.eigenvalue());<br />
           assertEquals(eigen.eigenvalue(i), property.eigenvalue());<br />
           assertEquals(eigen.property(eigen.eigenvalue(i)).algebraicMultiplicity(), property.algebraicMultiplicity());<br />
           assertEquals(2, property.algebraicMultiplicity());<br />
       }<br />
</i>

osigrtoeltParticipant 2010-12-16 at 10:09 am

Thanks, Haksun. How about complex eigenvalues?

For instance,

       Matrix A = new DenseMatrix(new double[][]{<br />
                    {1, -1},<br />
                    {0.5, 1}<br />
                });<br />

Haksun LiModerator 2010-12-16 at 10:17 am

It is the same except that

<br />
eigen.eigenvalue(i)<br />

returns a [tt:1r3zqrmz]Complex[/tt:1r3zqrmz] number.

E.g.,

<br />
        Matrix A = new DenseMatrix(new double[][]{<br />
                    {1, -1},<br />
                    {0.5, 1}<br />
                });<br />
        Eigen eigen = new Eigen(A);<br />
<br />
        String eigenvalue1 = "1-0.707107i";<br />
        String eigenvalue2 = "1+0.707107i";<br />
        assertTrue(eigen.eigenvalues().hasApproximately(1e-6, eigenvalue1, eigenvalue2));<br />
<br />
        assertEquals(2, eigen.size());<br />
        assertEquals(0, eigen.realEigenvalues().length);<br />
<br />
        for (int i = 0; i < eigen.size(); ++i) {<br />
            Number eigenvalue = eigen.eigenvalue(i);<br />
            Property property = eigen.property(eigenvalue);<br />
            assertEquals(eigenvalue, property.eigenvalue());<br />
            assertEquals(eigen.eigenvalue(i), property.eigenvalue());<br />
            assertEquals(eigen.property(eigen.eigenvalue(i)).algebraicMultiplicity(), property.algebraicMultiplicity());<br />
            assertEquals(1, property.algebraicMultiplicity());<br />
        }<br />