Generalising Sylvester’s Law of Inertia

Sylvester’s Law of Inertia is a statement about how the Eigenvalues of Hermitian matrices change under matrix *-congruence. Equivalently, it is a statement about possible forms of diagonal, normalised matrix representations of sesquilinear forms.

Using the former interpretation, it is possible to generalise the theorem a bit. This generalisation is in fact of practical importance. (This importance is not described here. Essentially it naturally arises because if the covariance matrix of a random vector X is R, then the covariance matrix of AX is ARA’, where a dash denotes conjugate-transpose).

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