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Structured orthogonal and unitary matrices: factorization, geometry and singularities

Ali Aliev

Palomar Registrypalomar/PALOMAR-2026-10-06-000011Oct 6, 2026math.DGscore 0

Block-diagonal matrix factors interleaved with fixed coordinate permutations define low-parametric matrix families. This formalization proves orthogonal-factor sufficiency for two and three real layers, its complex unitary analogue, and explicit four-layer counterexamples over both fields. It establishes an exact two-factor cell-rank criterion and real and unitary cosine-sine exhaustion for the (p,2,p) shuffle family. Two-layer images with complete within-layer block sharing, arbitrary overlap and arbitrary fixed carriers have a smooth atlas in the inherited matrix topology, an exact tangent…

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