A Result for Orthogonal Plus Rank-1 Matrices

TL;DR

This paper investigates the properties of matrices formed by adding an orthogonal matrix and a rank-1 outer product, establishing a relation between vector norms and singular values, with a key theorem providing the main result.

Contribution

It introduces a novel relation connecting vector norms and singular values for matrices composed of an orthogonal matrix plus a rank-1 outer product.

Findings

Established a relation between vector norms and singular values.

Proved a main theorem characterizing the matrix sum.

Provides insights into the structure of orthogonal plus rank-1 matrices.

Abstract

In this paper the sum of an orthogonal matrix and an outer product is studied, and a relation between the norms of the vectors forming the outer product and the singular values of the resulting matrix is presented. The main result may be found in Theorem 1.