This is continuation of previous post https://eikonal.wordpress.com/2010/02/17/some-examples-of-the-moore-penrose-inverse/
Example (added 2010.02.25 Fri): light-cone vector (i.e. ) () can be cast via Dirac matrices into the nilpotent (i.e. ) matrix , where (and ).
To find the MPI of that matrix, start with the transposed matrix . It is also nilpotent (i.e. ) but products following (singular) product are non-zero:
Now calculate via following limiting process:
Direct check verifies that all four defining properties of MPI are satisfied.
Note that corresponds to the vector , where is the space-parity operator.
The two products and are projectors: (), . They project on two light cone directions defined by vectors and .
More at this blog:
- Some examples of the Moore-Penrose inverse – https://eikonal.wordpress.com/2010/02/17/some-examples-of-the-moore-penrose-inverse/
- Weighted Moore-Penrose inverse – https://eikonal.wordpress.com/2010/02/19/weighted-moore-penrose-inverse/