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
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
To use the Java version:
To use the Java Web Start version:
- Make sure the Java Runtime Environment
is installed and that java is in your path.
- Click on the Java link above.
- Choose to save/download the file perplectic.jar.
- Change to the directory where the file was saved.
- Run the java code: java -jar perplectic.jar.
- Make sure Java Web Start
is installed and that javaws is in your path.
- Make sure your browser has been configured to use Java Web Start to open jnlp files.
- To set this up in Netscape/Mozilla:
- Go to Edit -> Preferences -> Helper Applications.
- Create a new type:
- MIME Type: application/x-java-jnlp-file
- Extension: jnlp
- Open it with: JRE_PATH/javaws/javaws
(JRE_PATH = path to Java Runtime Environment)
- To set this up for use with Windows/IE:
- Go to Control Panel -> Folder Options -> File Types.
- Create a New File Type:
- Extension: JNLP
- Advanced -> File Type: JNLP File
- Change the newly created File Type:
- Advanced -> New Action: Open
- Application used to perform action:
"C:\Program Files\Java Web Start\javaws.exe" "%1"
- Click on the Java Web Start link above and it should open Java Web Start.