Given a symmetric bilinear form, I have not yet understood how one can find:
- whether the BLF is positiv-definite
- the signature without calculating eigenvalues etc..
- check if the BLF is degenerate or not
- get an orthogonal basis for this BLF
Although I understand the concepts in theory, I would like to figure out a way to do the computations. Is there such an algorithm? From what I have seen, one can do the following to get the signature:
- reduce the matrix ($M$) that represents the BLF to a diagonal form and apply the same row and column operations to the standard matrix ($E$). When $M$ is in diagonal form, one can then easily read the signature from the diagonal entries.
Is there a name for this technique? How could I use this technique to check if the BLF is positiv definit and degenerate? I can't find any clear information about these topics. Any clarification would be greatly appreciated, as I have been struggling for days with these topics.
Thank you in advance.