We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on “transfer principles...

A section of a Riemannian G-manifold M is a closed submanifold Σ which meets each orbit orthogonally. It is shown that the algebra of G-invariant differential forms on M which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on Σ which are invariant with respect to the generalized Weyl group of Σ, un...

A b s t r a c t. The unbiased estimator of risk of the orthogonally invariant es-timator of the skew-symmetric normal mean matrix is obtained, and a class of minimax estimators and their order-preserving modification are proposed. The estimators have applications in paired comparisons model. A Monte Carlo study to compare the risks of the estimators is given.

The term “manifold of n dimensions” in this setting describes a set of n variables that independently take on the real values from −∞ to ∞ ([12], p 116). Motivated by this idea, one can assert that Euclidean geometry of E (Euclidean space) can be completely characterized by the invariants of the Euclidean group of transformations. As is well-known, this Lie group of (orientation-preserving) iso...

The expected number of real projective roots orthogonally invariant random homogeneous polynomial systems is known to be equal the square root Bézout number. A similar result for multi-homogeneous systems, through a product orthogonal groups. In this note, those results are generalized certain families sparse with no invariance assumed.

We study the set of circles which intersect a Dupin cyclide in at least two different points orthogonally. Dupin cyclides can be obtained by inverting a cylinder, or cone of revolution, or by inverting a torus. Since orthogonal intersection is invariant under Möbius transformations we first study the ortho-circles of cylinders/cones of revolution and tori and transfer the results afterwards.