  In this article we consider the following equivalence relation on the class
of all functions of two variables on a set $X$: we will say that $L,M: X\times
X\to \mathbb{C}$ are rescalings if there are non-vanishing functions $f,g$ on
$X$ such that $M\left(x,y\right)=f\left(x\right)g\left(y\right)
L\left(x,y\right)$, for any $x,y\in X$. We give criteria for being rescalings
when $X$ is a topological space, and $L$ and $M$ are separately continuous, or
when $X$ is a domain in $\mathbb{C}^{n}$ and $L$ and $M$ are
sesqui-holomorphic.
  A special case of interest is when $L$ and $M$ are symmetric, and $f=g$ only
has values $\pm 1$. This relation between $M$ and $L$ in the case when $X$ is
finite (and so $L$ and $M$ are square matrices) is known to be characterized by
the equality of the principal minors of these matrices. We extend this result
to the case when $X$ is infinite. As an application we characterize
restrictions of isometries of Hilbert spaces on weakly connected sets as the
maps that preserve the volumes of parallelepipeds spanned by finite collections
of vectors.
