Abstract
AbstractSum 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 them-dimensional complex column vector space mwith 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.
Highlights
Satisfying the three matrix equations AXA = A, XAX = X and AX = XA; the orthogonal projector onto M is a Hermitian idempotent matrix X, denoted by PM, that satis es range(X) = M; the collection of all orthogonal projectors of order m is denoted by COmP; a complex matrix U ∈ Cm×m is unitary if U∗U = UU∗ = Im; a Q ∈ Rm×m is call an orthogonal matrix if QTQ = QQT = Im
The aim of this paper is to provide a most detailed description on twists of two linear subspaces and two orthogonal projectors within the scope of linear algebra
It can be seen from Theorems 2.6(a) and 2.8(b) that all possible ordinary operations of two subspaces M and N in Cm can be decomposed as the direct sums of the following eight canonical subspaces: M ∩ N, M ∩ N⊥, M⊥ ∩ N, M⊥ ∩ N⊥, M
Summary
Substituting (2.41)–(2.48) into (2.33)–(2.40), we obtain a group of simultaneous direct sum decomposition identities as follows. (e) The following ve-term simultaneous direct sum decomposition identities hold (f) The following orthogonal projector decomposition identities hold
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