Dimensional analysis, first formalized by Vaschy and later Buckingham, enforces dimensional homogeneity and underpins classical scaling techniques in physics. While traditionally applied to the analytical analysis of differential equations and small algebraic systems, discussions of its extension to the linear algebra of systems of dimensioned equations have only recently gained traction—coinciding with the rapid growth of multiphysics simulation. In multiphysics settings, nonlinear and spatio-temporal couplings produce algebraic systems with heterogeneous physical dimensions whose physical dimensions and sparsity differ markedly from their monophysics counterparts. In fact, existing tools for assessing coupling strength, sensitivity, and solver stability are typically problem-specific or limited to small dimensionless examples, and thus do not capture how diverse physical units influence the applicability of standard algebraic techniques. In this work, we show that dimensional heterogeneity can invalidate normed-algebra arguments and spectral-radius bounds commonly used in the convergence analyses of iterative solvers, as many algebraic quantities in heterogeneously dimensioned matrices become fundamentally uncomparable. To address these algebraic inconsistencies, we introduce the Vaschy–Buckingham preconditioner, a simple scaling that converts a dimensionally heterogeneous multiphysics system into a homogeneous, dimensionless one, thereby restoring compatibility with conventional linear-algebraic analysis. Our results clarify the algebraic distinctions between mono- and multiphysics systems and provide a principled foundation for comparing coupling strengths across different physics in a unified framework.