Page 57 - Textos de Matemática Vol. 44

PARALLEL SUMS 47
2. Parallel sums
Let R1 and R2 be two resistors with (scalar) resistances r1 and r2. It is known from physics that if the resistors are connected in series the total resistance is r1 + r2 , and if the resistors are wired in parallel the total resistance is given by
(2.1) r1r2 =r1(r1 +r2)−1r2 =(1 + 1)−1. (r1 +r2) r1 r2
Assuming that r1 and r2 are positive numbers, the number (2.1) is called the parallel sum of r1 and r2, often denoted by r1 : r2. If one of them is zero and the other is positive we can handle the situation letting r1 : r2 = 0.
Extending this result to two resistive n-port networks N1, N2, with impe- dance matrices Z1, Z2, then the impedance matrix of the series and the parallel connections are, respectively,
(Z1 + Z2), Z1(Z1 + Z2)−1Z2
if Z1 and Z2 are nonsingular.
Since nonnegative definite matrices are a generalization of nonnegative sca-
lars, a kind of extension of r1 : r2 leads to the definition of parallel sum of matrices.
Definition 2.1. Let A,B ∈ Cn×n be nonsingular and nonnegative definite. The parallel sum of A and B, denoted by A : B, is defined by
or by
A : B = A(A + B)−1B, A : B = (A−1 + B−1)−1.
It is clear that for resistive networks the impedance matrix is nonnegative definite and hermitian. This is the case for many physical and engineering ap- plications. If A and B are nonsingular hermitian nonnegative definite matrices, then that A : B is also hermitian nonnegative definite is just a consequence of the definition.
In the following an important fact established on hermitian PSD matrices will be used, that is, if A and B are hermitian and PSD then R(A + B) = R(A) + R(B).
When (A + B) is singular the parallel addition may still be defined as in [5], that is, replacing (A + B)−1 by a generalized inverse of (A + B).
Definition 2.2. For a pair of matrices A and B of the same order the parallel sum of A and B, denoted by A∓B, is defined by
(2.2) A ∓ B = A(A + B)−B,