  Inspired by recent work in the theory of central projections onto
hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those
with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also
prove that every Gorenstein perfect algebra of grade 1 can be presented, as a
module, by a symmetric matrix. Both results are derived from the same
elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor,
a symmetric syzygy matrix. In addition, we establish a correspondence, roughly
speaking, between Gorenstein perfect algebras of grade 1 that are birational
onto their image, on the one hand, and self-linked perfect ideals of grade 2
that have one of the self-linking elements contained in the second symbolic
power, on the other hand. Finally, we provide another characterization of these
ideals in terms of their symbolic Rees algebras, and we prove a criterion for
these algebras to be normal.
