There is a substantial literature on variational principles and estimates for eigenvalues of selfadjoint transformations in inner product spaces, most of which depends upon the fact that the associated quadratic forms are real and that the real numbers are totally ordered. Here we show that by using a partial ordering in the complex numbers, defined by means of a complex cone, variational principles and estimates can be obtained for certain classes of normal transformations. We will assume in what follows that X is a finite-dimensional inner product space over the complex numbers, with (., *) denoting the inner product, and that N is a normal transformation in X, i.e. NN* = N*N. The extensions of the results below to an infinitedimensional Hilbert space are straightforward. We will assume that N has eigenvalues X1, X2, * * * , XI (with Re Xi+,1_ Re Xi) counting multiplicity, and associated orthogonal eigenvectors wi. Then N can be written as N= EXiEi, with Ei the orthogonal projection on the span of wi. Let pti, .2, * * * , Ak (with Re Ai+, _ Re pi) be the eigenvalues, not counting multiplicity. We will assume that