On relationships between two linear subspaces and two orthogonal projectors

• Main Author: Tian Yongge
• Format: Article
• Language: English
• Published: De Gruyter 2019-10-01
• Series: Special Matrices
• Subjects: linear subspace; orthogonal complement; sum; intersection; orthogonal projector; matrix decomposition; moore–penrose inverse; generic position; commutativity; norm; ep matrix; 15a03; 15a09; 15a24; 15a27; 15b57; 47a15; 47h05
• Online Access: https://doi.org/10.1515/spma-2019-0013

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m-dimensional complex column vector space 𝔺m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.