Two symmetric matrices underlie our understanding of microevolutionary change. The first is the matrix of nonlinear selection gradients (γ) which describes the individual fitness surface. The second is the genetic variance–covariance matrix (G) that influences the multivariate response to selection. A common approach to the empirical analysis of these matrices is the element-by-element testing of significance, and subsequent biological interpretation of pattern based on these univariate and bivariate parameters. Here, I show why this approach is likely to misrepresent the genetic basis of quantitative traits, and the selection acting on them in many cases. Diagonalization of square matrices is a fundamental aspect of many of the multivariate statistical techniques used by biologists. Applying this, and other related approaches, to the analysis of the structure of γ and G matrices, gives greater insight into the form and strength of nonlinear selection, and the availability of genetic variance for multiple traits.