Inconsistency Removal of Reduced Bases in Parametric Model Order Reduction by Matrix Interpolation using Adaptive Sampling and Clustering

The paper defines the same reference-basis alignment and transformation T_k=(R^T V_k)^{-1} after constructing R via an SVD of the library of sampled bases and enforces Ṽ_k^T R = I; it then applies the congruence transforms to obtain transformed operators for interpolation (their eqs. (6)–(9) as described in the text) . It also defines principal subspace angles through the SVD of V_i^T V_j (their eqs. (11)–(12)) and notes that mode switching/truncation can render R^T V_k singular—i.e., a 90° principal angle—making T_k singular . The candidate solution gives the precise equivalence “T_k exists iff all principal angles < 90°,” and explains conditioning via σ_min, which is fully consistent with the paper’s framework. One minor discrepancy: the paper’s printed “approximate relative H2 error” expression (eq. (13) as typeset) omits the square magnitude in the integrand; the candidate’s H2 definition is the standard one and is correct, so this appears to be a benign typographical slip in the paper .