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https://issues.apache.org/jira/browse/MATH-416?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=12909140#action_12909140
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Dimitri Pourbaix commented on MATH-416:
---------------------------------------
Returning V such that det(V) is always +1 would please you but would cost some
substantial computation in general, which would make mad users for whom speed
is an issue. A compromise could consist in a second accessor to V (and to Vt
too I guess) which would guarantee that the determinant is +1. Would it be OK
for you?
> EigenDecompositionImpl.getV() returns eigen matrix with indeterminate
> determinant
> ---------------------------------------------------------------------------------
>
> 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
> Assignee: Dimitri Pourbaix
>
> 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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