On the product of oblique projectors

Groβ, Jürgen; Trenkler, Götz

doi:10.1080/03081089808818561

Cited by 15 publications

(5 citation statements)

References 2 publications

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“…We are especially interested in the situation of Theorem 1, that is, the case where the composition of boundary problems corresponds to the composition of their generalized Green's operators. For testing when G 2 G 1 is an outer inverse of T 1 T 2 , we use the following characterization from [18,13], which is based on results from [24] and [25]. It gives necessary and sufficient conditions on the subspaces B ⊥ 1 , T 2 (B ⊥ 2 ), E 2 , and T −1 1 (E 1 ) such that the revers order law…”

Section: Algorithm 1 (Composition)mentioning

confidence: 99%

Composing and Factoring Generalized Green’s Operators and Ordinary Boundary Problems

Korporal

Regensburger

2014

Algebraic and Algorithmic Aspects of Differential and Integral Operators

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We consider solution operators of linear ordinary boundary problems with "too many" boundary conditions, which are not always solvable. These generalized Green's operators are a certain kind of generalized inverses of differential operators. We answer the question when the product of two generalized Green's operators is again a generalized Green's operator for the product of the corresponding differential operators and which boundary problem it solves. Moreover, we show thatprovided a factorization of the underlying differential operator-a generalized boundary problem can be factored into lower order problems corresponding to a factorization of the respective Green's operators. We illustrate our results by examples using the Maple package IntDiffOp, where the presented algorithms are implemented.

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Section: Algorithm 1 (Composition)mentioning

confidence: 99%

Composing and Factoring Generalized Green’s Operators and Ordinary Boundary Problems

Korporal

Regensburger

2014

Algebraic and Algorithmic Aspects of Differential and Integral Operators

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“…The first of the following necessary and sufficient conditions for the product of P and Q to be a projector is mentioned as an exercise without proof in [3, p. 339]. In [11,Lemma 3] the same result is formulated for matrices but the proof is valid for arbitrary vector spaces. The second necessary and sufficient condition for the matrix case is given in [28, Lemma 2.2].…”

Section: Products Of Projectorsmentioning

confidence: 99%

“…In [11,Lemma 3] the same result is formulated for matrices but the proof is valid for arbitrary vector spaces. The second necessary and sufficient condition for the matrix case is given in [28,Lemma 2.2].…”

Section: Products Of Projectorsmentioning

confidence: 99%

“…In general, these conditions are necessary but not sufficient for commutativity of P and Q, see [11,Ex. 1].…”

Section: Lemma 6 the Composition P Q Is A Projector If And Only Ifmentioning

confidence: 99%

“…The validity of the reverse order law can be reduced to the question whether the product of two projectors is a projector (Section 2). This problem is studied in [11,25,28] for finite-dimensional vector spaces. We discuss necessary and sufficient conditions that carry over to arbitrary vector spaces and can be expressed in terms of the kernels and images of the respective operators alone (Section 4).…”

Section: Introductionmentioning

confidence: 99%

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On the product of projectors and generalized inverses

Korporal

Regensburger

2013

Linear and Multilinear Algebra

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We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. This problem is equivalent to the question when the product of two projectors is again a projector, and we discuss necessary and sufficient conditions in terms of their kernels and images alone. We give a new representation of the product of generalized inverses that does not require explicit knowledge of the factors. Our approach is based on implicit representations of subspaces via their orthogonals in the dual space. For Fredholm operators, the corresponding computations reduce to finite-dimensional problems. We illustrate our results with examples for matrices and linear ordinary boundary problems.

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On nested block designs geometry

Kala

2009

Stat Papers

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On the product of oblique projectors

Cited by 15 publications

References 2 publications

Composing and Factoring Generalized Green’s Operators and Ordinary Boundary Problems

Composing and Factoring Generalized Green’s Operators and Ordinary Boundary Problems

On the product of projectors and generalized inverses

On nested block designs geometry

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