the signature of a metric is often defined to be the number of positive eigenvalues minus negative eigenvalues of the metric. this definition has always seemed a little suspicious to me. eigenvalues are defined for endomorphisms of a linear space, whereas the metric is a bilinear functional on the vector space. it is not clear to me how one would write down an eigenvalue equation for the metric. insofar as the metric can always be written as a matrix in local coordinates, i guess you can just shutup and pretend that it is a linear transformation, and calculate its eigenvalues, but it seems to me to be a suspicious procedure. any thoughts?