Perplectic Eigenproblems

An interactive supplement to:

D. Steven Mackey, Niloufer Mackey and Daniel M. Dunlavy,
Structure preserving algorithms for perplectic eigenproblems,
Electronic Journal of Linear Algebra, 13:10-39, 2005.


These animations show the progression of a structure-preserving "row-cyclic" sweep on a matrix, as described in Section 8.   The initial matrix has symmetry or skew-symmetry about the main diagonal as well as the main anti-diagonal.  These pictures display a scheme for selecting a sequence of 4x4 submatrices that will also be symmetric or skew-symmetric about both diagonals, with the property that at the end of the sweep, all parts of the matrix will have been visited.

Dark purple entries represent the current target submatrix.  Light purple entries represent the entries that have been part of a target submatrix in previous iterations.  Grey entries represent the entries that have not yet been part of a target submatrix.  At the end of a sweep, every entry of the matrix is some shade of purple, showing that all entries of the matrix are reachable.

When the matrix is of odd size, the sweep also involves 3x3 subproblems.  These subproblems are represented by blue entries.

Symmetric Persymmetric Examples

These pictures show the structure-preserving algorithm in action on a matrix that is symmetric about both diagonals.  In this case the structured canonical form is an "X" (see Section 7).    The colors represent the magnitudes of the entries, using the color scale shown in each example. The color scale is logarithmic with reds representing entries with the largest magnitudes (on the order of 1), ranging through yellows (1e-5) and light blues (1e-10) to dark blues representing those with the smallest magnitudes (1e-15). Finally, black represents entries with magnitudes below machine epsilon (about 2.204e-16).

Compatibility Notes

The different formats of the perplectic sweepshows listed above have been tested using the Windows 2000 Professional (using Internet Explorer 6.0 and Firefox 1.0 browsers) and RedHat Linux 9.0 (using Mozilla 1.6 browser) operating systems. Below is a table listing the helper applications (along with version numbers) that were used in testing the various formats.

Number Windows (IE 6.0; Firefox 1.0) Linux (Mozilla 1.6)
1 MS Office (2003) OpenOffice (1.0)
2 Windows Media Player (9.0) MPlayer (1.0pre6-3.2.2);   Mozplugger (1.7.1)
3 Macromedia Flash Player (7.0) Macromedia Flash Player (6.0)
4 Java Runtime Environment (1.4.1_01) Java Runtime Environment (1.4.2_02)
5 Java Web Start (1.4.1_01) Java Web Start (1.4.2_02)

To use the Java version:
  1. Make sure the Java Runtime Environment is installed and that java is in your path.
  2. Click on the Java link above.
  3. Choose to save/download the file perplectic.jar.
  4. Change to the directory where the file was saved.
  5. Run the java code: java -jar perplectic.jar.
To use the Java Web Start version:
  1. Make sure Java Web Start is installed and that javaws is in your path.
  2. Make sure your browser has been configured to use Java Web Start to open jnlp files.
  3. Click on the Java Web Start link above and it should open Java Web Start.