### Abstract:

The fixed-point iteration is a simple method for finding the maximal
Hermitian positive
definite solutions of the matrix equations $X\pm A^*X^{-1}A=I$
(the plus/minus equations).
The convergence
of this method
may be very slow if the initial matrix is not chosen carefully. A strategy
for choosing
better initial matrices has been recently proposed by Ivanov, Hasanov and
Uhlig. They
proved that this strategy can improve the convergence in general and observed
from numerical
experiments that dramatic improvement happens for the plus equation
with some matrices $A$. It turns
out that
the matrices $A$ are normal for those examples. In this note
we prove a result
that explains the dramatic improvement in convergence for normal
(and thus nearly
normal) matrices for the plus equation. A similar result is also proved
for the minus
equation.