The Scaling-Rotation Chain Metric for Positive-Definite Two by Two Matrices

The scaling-rotation distance between positive-definite matrices minimizes the distance between their eigendecomposition fibers. Its associated chain metric also allows changes of eigenframe at repeated eigenvalues. For real two by two matrices we compute that chain metric explicitly. Writing \(m\) for the normalized logarithmic determinant, \(r\) and \(s\) for the normalized logarithmic spectral gaps, and \(\alpha\) for the acute angle between the top eigenlines, its square is \((m_Y-m_X)^2+\min\{(r-s)^2+k\alpha^2,(r+s)^2\}\). We characterize all minimizing paths in the finite-piece class and determine exactly when a minimal smooth scaling-rotation curve remains globally shortest, including the orthogonal-eigenline exception. We also prove determinant splitting and a sufficient spectral-gap condition in arbitrary dimension. These calculations use the established Groisser–Jung–Schwartzman framework. They answer the two-dimensional case of Jung’s Oberwolfach question, not its general-dimensional classification. This is an unrefereed preprint with AI assistance disclosed; no absolute priority, independent review or formal verification is claimed. Possible overlap with the original authors’ separately listed existence manuscript is not excluded.