Principal Values and Principal Subspaces of Two Subspaces of Vector Spaces with Inner Product

TL;DR

This paper investigates the angles between subspaces in Euclidean space, introduces principal values and subspaces via eigenanalysis, and expands classical results with new geometric and algebraic insights.

Contribution

It introduces principal values and subspaces using eigenvalues and eigenvectors, providing a new geometric interpretation and canonical representation of subspace relationships.

Findings

Angles between subspaces equal angles between their orthogonal complements

Principal values and subspaces are characterized via eigenanalysis

Canonical matrix exhibits duality properties

Abstract

In this paper is studied the problem concerning the angle between two subspaces of arbitrary dimensions in Euclidean space $E_{n}$. It is proven that the angle between two subspaces is equal to the angle between their orthogonal subspaces. Using the eigenvalues and eigenvectors of corresponding matrix representations, there are introduced principal values and principal subspaces. Their geometrical interpretation is also given together with the canonical representation of the two subspaces. The canonical matrix for the two subspaces is introduced and its properties of duality are obtained. Here obtained results expand the classic results given in [1,2].