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From "Tom Milac (JIRA)" <j...@apache.org>
Subject [jira] Created: (MATH-416) EigenDecompositionImpl.getV() returns eigen matrix with indeterminate determinant
Date Mon, 13 Sep 2010 23:21:33 GMT
```EigenDecompositionImpl.getV() returns eigen matrix with indeterminate determinant
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Key: MATH-416
URL: https://issues.apache.org/jira/browse/MATH-416
Project: Commons Math
Issue Type: Improvement
Affects Versions: 2.1
Environment: Mac OS X 10.6.4
Reporter: Tom Milac

A call to EigenDecompositionImpl.getV() returns a RealMatrix the columns of which are the
eigenvectors of the matrix with which EigenDecompositionImpl is constructed. Because EigenDecompositionImpl
works only with real, symmetric matrices, the eigenvectors (columns) returned are orthogonal.
In addition, the eigenvectors are normalized to have 2-norm = 1. Unfortunately, for 3x3 input
matrices, the determinant of the eigenvector matrix is indeterminate, sometimes +1 and other
times -1.  The -1 output can be
'repaired' simply by multiplying the matrix by -1.  Example code is included below.

Because the columns are eigenvectors, the result with either determinant is correct.  However,
in the case that the matrix returned is to be interpreted as specifying a coordinate system,
the principal axes of a body in my case,  the +1 result specifies a right-handed coordinate
for the principal coordinate system of the body, and the -1 result specifies a left-handed
coordinate system.  Once discovered, this indeterminacy is easy to deal with, but nevertheless
an inconvenience.

I believe it would improve EigenDecompositionImpl.getV() to return an eigenvector matrix with
a consistent determinant = +1.

Tom Milac

---------------------------------------------------------

import org.apache.commons.math.geometry.NotARotationMatrixException;
import org.apache.commons.math.geometry.Rotation;
import org.apache.commons.math.linear.Array2DRowRealMatrix;
import org.apache.commons.math.linear.EigenDecompositionImpl;
import org.apache.commons.math.linear.InvalidMatrixException;
import org.apache.commons.math.linear.LUDecompositionImpl;
import org.apache.commons.math.linear.RealMatrix;

/**
*
* @author Tom Milac
*/
public class BugReport {

/**
* Moment of inertia tensor #1.
*/
public static final double[][] MOI1 =
{{128.52722633757742, -29.11849805467669, 8.577081342861376},
{-29.11849805467669, 521.3276639228706,  35.512665035385666},
{8.577081342861376,  35.512665035385666, 490.2479495932442}};

/**
* Moment of inertia tensor #2.
*/
public static final double[][] MOI2 =
{{440.09350934414175, 44.23154125186637, -9.41455073681743},
{44.23154125186637,  387.1291457565648, -38.07596950448303},
{-9.41455073681743, -38.07596950448303, 762.0451513430822}};

/**
* Constructor.
*/
public BugReport() {
}

/**
* Main.
*/
public static void main(String[] args) {

// Compute the principal axes (eigenvectors) of the #1 moment
// of inertia tensor.
RealMatrix moi1  = new Array2DRowRealMatrix(MOI1);
RealMatrix axes1 = null;
EigenDecompositionImpl eigenDecompositionImpl = null;
try {
eigenDecompositionImpl = new EigenDecompositionImpl(moi1, 0.0d);

axes1 = eigenDecompositionImpl.getV();
} catch (InvalidMatrixException ex) {
System.err.println("MOI1: InvalidMatrixException thrown.");
System.err.println("Exiting ...");
System.exit(-1);
}

// Compute the principal axes (eigenvectors) of the #2 moment
// of inertia tensor.
RealMatrix moi2  = new Array2DRowRealMatrix(MOI2);
RealMatrix axes2 = null;
try {
eigenDecompositionImpl = new EigenDecompositionImpl(moi2, 0.0d);

axes2 = eigenDecompositionImpl.getV();
} catch (InvalidMatrixException ex) {
System.err.println("MOI2: InvalidMatrixException thrown.");
System.err.println("Exiting ...");
System.exit(-1);
}

// Determinant of axes 1 eigenvector matrix = -1.  If the matrix
// is interpreted as a Rotation, throws and Exception.
System.out.print("Determinant of the #1 moment of inertia tensor = ");
System.out.println(new LUDecompositionImpl(axes1).getDeterminant());

try {
Rotation axes1_rotation = new Rotation(axes1.getData(), 1.0E-7);
} catch (NotARotationMatrixException ex) {
System.out.println("NotARotationMatrixException thrown for 'axes1'.");
}

System.out.println();

// Determinant of axes 2 eigenvector matrix = +1.
System.out.print("Determinant of the #2 moment of inertia tensor = ");
System.out.println(new LUDecompositionImpl(axes2).getDeterminant());

try {
Rotation axes2_rotation = new Rotation(axes2.getData(), 1.0E-7);
} catch (NotARotationMatrixException ex) {
System.out.println("NotARotationMatrixException thrown for 'axes2'.");
}
}
}

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