scholarly journals Characterization of best approximations in normed linear spaces of matrices by elements of finite-dimensional linear subspaces

1981 ◽  
Vol 35 ◽  
pp. 109-120 ◽  
Author(s):  
K.K. Lau ◽  
W.O.J. Riha
Keyword(s):  
Best Approximations ◽  
Linear Spaces ◽  
Linear Subspaces ◽  
Normed Linear Spaces ◽  
Finite Dimensional

scholarly journals On relationships between two linear subspaces and two orthogonal projectors

2019 ◽  
Vol 7 (1) ◽  
pp. 142-212 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Vector Space ◽  
Matrix Decomposition ◽  
Complex Vector ◽  
Linear Subspaces ◽  
Survey Paper ◽  
Finite Dimensional ◽  
Generic Position ◽  
Ep Matrix ◽  
Orthogonal Projectors

Abstract 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.

scholarly journals Extension with larger norm and separation with double support in normed linear spaces

1980 ◽  
Vol 21 (1) ◽  
pp. 93-105 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Sets ◽  
Separation Theorems ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Linear Spaces ◽  
Double Support ◽  
Linear Subspaces ◽  
Normed Linear Spaces ◽  
Whole Space

We prove, in normed linear spaces, the existence of extensions of continuous linear functionals from linear subspaces to the whole space, with arbitrarily prescribed larger norm. Also, we prove that under an additional boundedness assumption, in the known separation theorems for convex sets, there exist hyperplanes which separate and support both sets.

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