Symmetric closed-form polar factor for the contrast subspace and a polar decomposition derivation of Helmert matrices

This preprint addresses the construction of orthogonal bases associated with the contrast subspace orthogonal to the all-ones vector, a setting that appears naturally in a wide range of statistical and algebraic contexts. The manuscript adopts a matrix-analytic viewpoint based on polar decomposition in order to compare different ways of parametrising this subspace. The main contribution is the introduction of a symmetric generating matrix whose orthogonal polar factor can be written explicitly in closed form. This construction yields an order-free orthonormal representation of the contrast subspace that is uniquely determined by the structure of the generator and does not rely on sequential orthogonalisation. The resulting orthogonal matrix may be interpreted as an intrinsic representative associated with the decomposition of the ambient space into the span of the all-ones vector and its orthogonal complement. Within the same framework, the classical Helmert matrix is reinterpreted as the polar factor of a structured triangular generator. This perspective provides an analytic derivation valid in arbitrary dimension and explains the hierarchical organisation of Helmert contrasts as a direct consequence of the generator’s structure. Several examples are included to illustrate how different generators induce different orthonormal parametrisations of the same subspace. The preprint is intended as a conceptual and structural contribution. Its purpose is to clarify the relationship between matrix structure, polar decomposition, and orthogonal parametrisations, rather than to address computational or numerical aspects.