Browne and Shapiro (1989) give several corollaries which establish properties of reflector matrices implied by various types of scale invariances. Their results include (Shapiro & Browne, 1989, page 8).

**Corollary 1.1 ***If ***S**(**q**)* is invariant under **G**1***,
then the sum of the diagonal elements of is zero.*

**Corollary 1.2 ***If* **S**(**q**)* is invariant under G2*,
then all diagonal elements of are zero.*

These corollaries provide convenient devices
for falsifying the dual assertion that a minimum has been obtained for
a given discrepancy function, *and* that the fitted model possesses
an invariance property.

Specifically, for a given discrepancy function,
the reflector matrix is computed after convergence. If the trace of the
reflector matrix is not zero, then *either a minimum has not been obtained,
or the model is not invariant under a constant scaling factor, or both*.
Then the individual diagonal elements of the reflector matrix are examined.
If they are not all zero, one can conclude that *either a minimum has
not been obtained, or the model is not invariant under changes of scale,
or both.*

From a practical standpoint, one must remember
issues of machine precision. One would only expect the above criteria
to be met to an acceptable level of machine precision. Consequently, *SEPATH*,
besides printing the reflector matrix, also reports (1) the trace of the
reflector matrix, and (2) the largest absolute value on the diagonal of
the reflector matrix.