and
The similarity transformation is B A Btr, where A is a symmetric matrix and B is an ortho-normal matrix. However, the routine for this product often is employed for matrices A and B that do not posses the aforementioned symmetry properties. It is left as an exercise for the reader to verify – or call to my attention any errors in – this table. The code-numbers as those actually employed within the library to represent the indicated symmetries.
|
|
|
Zero |
Identity |
Diagonal |
Ortho-normal |
Skew-symmetric |
Symmetric |
General |
|
|
B \ A |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
|
Zero |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
|
Identity |
5 |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
|
Diagonal |
4 |
6 |
4 |
4 |
0 |
2 |
1 |
0 |
|
Ortho-normal |
3 |
6 |
5 |
1 |
3 |
2 |
1 |
0 |
|
Skew-symmetric |
2 |
6 |
1 |
1 |
0 |
2 |
1 |
0 |
|
Symmetric |
1 |
6 |
1 |
1 |
0 |
2 |
1 |
0 |
|
General |
0 |
6 |
0 |
0 |
0 |
0 |
0 |
0 |
There are certain other routines that have such symmetry tables. For instance, the addition of two matrices:
|
|
|
Zero |
Identity |
Diagonal |
Ortho-normal |
Skew-symmetric |
Symmetric |
General |
|
|
+ |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
|
Zero |
6 |
6 |
5 |
4 |
3 |
2 |
1 |
0 |
|
Identity |
5 |
5 |
4 |
4 |
0 |
0 |
1 |
0 |
|
Diagonal |
4 |
4 |
4 |
4 |
0 |
0 |
1 |
0 |
|
Ortho-normal |
3 |
3 |
0 |
0 |
0 |
0 |
0 |
0 |
|
Skew-symmetric |
2 |
2 |
0 |
0 |
0 |
2 |
0 |
0 |
|
Symmetric |
1 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
|
General |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Copyright (c) 2003, 4 by R.I. ‘Scibor-Marchocki. Last revised Sunday 20-th June 2004.