  In this note we present an alternative way to obtain optimal interpolation
operators for two-grid methods applied to Hermitian positive definite linear
systems. Falgout and Vassilevski in [SIAM J. Numer. Anal, 42 (2004), pp.
1669-1693] and Zikatanov [Numer. Linear Algebra Appl., 15 (2008), pp. 439-454]
have characterized the $A$-norm of the error propagation operator of algebraic
multigrid methods. These results have been recently used by Xu and Zikatanov
[Acta Numer., 26 (2017), pp. 591-721] and Brannick, Cao et al. [SIAM J. Sci.
Comp, 40 (2018), pp. 591-721] to determine optimal interpolation operators.
Here we use a characterization not of the $A$-norm but of the spectrum of the
error propagation operator of two-grid methods, which was proved by Garc\'ia
Ramos, Nabben and Kehl and holds for arbitrary matrices. For Hermitian positive
definite systems this result leads to optimal interpolation operators with
respect to the $A$-norm in a short way, moreover, it also leads to optimal
interpolation operators with respect to the spectral radius. For the symmetric
two-grid method (with pre- and post-smoothing) the optimal interpolation
operators are the same. But for a two-grid method with only post-smoothing the
optimal interpolations (and hence the optimal algebraic multigrid methods) can
be different. Moreover, using the characterization of the spectrum, we can show
that the found optimal interpolation operators are also optimal with respect to
the condition number of the multigrid preconditioned system.
