On the Canonical Reduction of Quadratic Forms
Abstract
The standard method for simultaneously reducing two quadratic forms in n variables to sums of squares depends upon finding a set of n independent eigenvectors. The same method is applicable to the reduction of a hermitian matrix to diagonal form by a unitary matrix, as is done in quantum mechanics. This method is based on the following fundamental theorem concerning eigenvectors: If A and B are two hermitian matrices of order n, A being positive definite, and if λi is a k-fold root of the secular equation |B-λA|=0, then the equation (B-λiA)ξ=0 has k independent solutions. In other words, there are k independent eigenvectors corresponding to a k-fold root of the secular equation. A proof is given for this theorem which is concise and at the same time quite elementary.
- Publication:
-
Physical Review
- Pub Date:
- April 1949
- DOI:
- 10.1103/PhysRev.75.1088
- Bibcode:
- 1949PhRv...75.1088H