Parallel Subtraction of Matrices

Abstract

A new Hermitian semidefinite matrix operation is studied. This operatioin-called parallel subtraction-is developed from the theory of parallel addition. Since the theory of parallel addition is motivated by the analysis of interconnected electrical networks, parallel subraction may be interpreted in terms of the synthesis of electrical networks. The idea of subtraction is also extended to hybrid addition.

Bilateral shorted operators and parallel sums

“…Given two operators A, C e L(^i, ^2), it seems natural to study the existence of a solution of the equation A : X = C, that is, if there exists an operator B e L(Jfj, .Xf2} parallel summable with A such that A : B = C. For positive operators this question has been studied, for example, in [3,38,4,37], Clearly, equation Remark 5.15. Note that, according to our definition, it holds that C 4-A = C : (-A); in partic ular, several properties of parallel sum are inherited by parallel subtraction.…”

“…Given two operators A, C ∈ L(H 1 , H 2 ), it seems natural to study the existence of a solution of the equation A : X = C, that is, if there exists an operator B ∈ L(H 1 , H 2 ) parallel summable with A such that A : B = C. For positive operators this question has been studied, for example, in [3], [39], [4] and [38]. Clearly, equation A : X = C may have no solutions for some pair of operators (A, C).…”

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

“…In fact, the study of electrical circuits motivated the definition of the parallel addition and parallel subtraction operators, which corresponds to the inf-convolution and deconvolution of quadratic functions. Anderson [6,7,8] defined the parallel addition operator, and Mazure [174,177,175,178,176,119] studied its properties from a Convex Analysis perspective (some of her results also apply to nonconvex functions). Consider [117, Example IV.2.3.8 p. 165]: an electrical circuit is made up of two generalized resistors A 1 and A 2 connected in parallel, and we want to find the equivalent resistor.…”

Proyecciones oblicuas y complementos de Schur

“…Corolario 6.3.10. Si A, B ∈ L(H 1 , H 2 ) son sumables en paralelo, entonces R(A : B) = R(A) ∩ R(B).Resta paralelaDados dos operadores A, C ∈ L(H 1 , H 2 ), parece natural estudiar la existencia de una solución de la ecuación A : X = C, es decir, si existe un operador B ∈ L(H 1 , H 2 ) paralelamente sumable con A tal que A : B = C. Para operadores positivo, esta pregunta ha sido estudiada, entre otros, por Anderson, Duffin y Trapp en[2], Anderson Morley y Trapp en[106], y por Pekarev en[105]. Claramente, la ecuación A : X = C puede no admitir solución.…”