On Grioli's minimum property and its relation to Cauchy's polar decomposition

TL;DR

This paper explores Grioli's 1940 note demonstrating the Frobenius norm minimization property of the polar factor in matrix decompositions, linking classical results to geometric interpretations.

Contribution

It provides a detailed translation and commentary on Grioli's original work, highlighting its significance in understanding the optimality of the polar decomposition.

Findings

Grioli's note confirms the Frobenius norm minimization property in 3D.

The work connects the polar factor's optimality to geometric and displacement concepts.

It clarifies the historical development of the polar decomposition's optimality properties.

Abstract

The unitary polar factor of a matrix F is the unitary matrix Q realizing the minimum of the norm of F-Q over all unitary matrices Q. Tracing back the development on the optimality of the polar factor to its presumable roots, in this paper we present a commented translation of a note by G. Grioli from 1940 (Boll.Un.Math.Ital.2,452-455(1940)) showing the minimization property for the Frobenius norm in dimension 3; in his words: "We show that for the homogeneous displacement, tangent to any finite displacement, there exists a minimum property completely analogous to a well known property valid for infinitesimal displacements." Keywords: polar decomposition, optimality of the polar factor, Euclidean distance, geodesic distance, Euclidean movement