Geometric Residual Projection in Linear Regression: Rank-Aware Operators and a Geometric Multicollinearity Index

Residuals play a central role in linear regression, yet their geometry is often hidden by inverse- and pseudoinverse-based formulas. We develop a rank-aware framework for residual projection that makes the underlying orthogonality explicit. When the design matrix has codimension one, the unexplained component of the response lies along a single unit normal to the predictor space, and the residual projector reduces to the rank-one operator nn⊤, avoiding matrix inversion. For general designs, the residual lies in a higher-dimensional orthogonal complement spanned by an orthonormal basis N, and the residual projector factorizes as NN⊤. Using cross-products, wedge products, and Gram determinants, we provide basis-independent characterizations of the residual subspace. We further introduce the Geometric Multicollinearity Index (GMI), a scale-invariant diagnostic derived from the polar sine that quantifies the collapse of predictor-space volume under multicollinearity. Synthetic perturbation studies and an illustrative real-data experiment show that the proposed projectors reproduce ordinary least squares residuals, that GMI responds predictably to controlled collinearity, and that the projector viewpoint clarifies the distinction between regression residuals and PCA reconstruction residuals in both full-rank and rank-deficient settings.