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 ${\ensuremath{\lambda}}_{i}$ is a $k$-fold root of the secular equation $|B\ensuremath{-}\ensuremath{\lambda}A|=0$, then the equation $(B\ensuremath{-}{\ensuremath{\lambda}}_{i}A)\ensuremath{\xi}=0$ 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.