On the sensitivity of the canonical angles of a unitoid matrix

A unitoid matrix is a square complex matrix that can be brought to diagonal form by a Hermitian congruence transformation. The canonical angles of a nonsingular unitoid matrix $A$ are (up to the factor $1/2$) the arguments of the eigenvalues of the cosquare of $A$, which is the matrix $A^{-*}A$. We derive an estimate for the derivative of an eigenvalue of the cosquare in the direction of the perturbation in $A^{-*}A$ caused by a perturbation in $A$.