Orthogonal projectors onto spaces of periodic splines

In the paper subadditive measure on the lattice of orthogonal projectors of von Neumann algebra is considered. The basic peoperties of the subadditive measure are estabelished and proved.

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Sum of Orthogonal Projectors

Matrix Tricks for Linear Statistical Models ◽

10.1007/978-3-642-10473-2_8 ◽

2011 ◽

pp. 151-154

Author(s):

Simo Puntanen ◽

George P. H. Styan ◽

Jarkko Isotalo

Keyword(s):

Orthogonal Projectors

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Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case

Axioms ◽

10.3390/axioms8020056 ◽

2019 ◽

Vol 8 (2) ◽

pp. 56 ◽

Cited By ~ 2

Author(s):

Galina Kurina

Keyword(s):

Differential Equations ◽

Asymptotic Solution ◽

Initial Value Problems ◽

Singular Matrix ◽

Singularly Perturbed Systems ◽

English Transl ◽

Initial Value ◽

Boundary Functions ◽

Linear Differential ◽

Orthogonal Projectors

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) ′ (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

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On relationships between two linear subspaces and two orthogonal projectors

Special Matrices ◽

10.1515/spma-2019-0013 ◽

2019 ◽

Vol 7 (1) ◽

pp. 142-212 ◽

Cited By ~ 2

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.

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