This post is a slight modification of section 2.0 “The Parallel Sum of Vectors” from W. N. Anderson & G. E. Trapp (1987) “The harmonic and geometric mean of vectors”, Linear and Multilinear Algebra, 22:2, 199-210.
We will consider vectors in a real N dimensional inner product space, although some of the results given herein apply equally well to the complex case.
We use a, b, . . . to represent vectors. The inner product of a and b is written ⟨a, b⟩ and the norm of a vector a is given by ||a|| = ⟨a, a⟩1/2.
Given vectors a and b, their arithmetic mean is defined by (a + b)/2. To define the harmonic mean we require an involution which is described next.
If a is a non-zero vector, define a′ (to represent the reciprocal of a) by
a′ = a/||a||² = (1/||a||)(a/||a||) = â/||a||.
First notice that for a non-zero scalar k,
(ka)‘ = ka / ||ka||² = a / k||a||² = a′/k.
Next we see that a″ = (a′)′ = (a/||a||²)′ = ||a||²a′ = a and therefore ′ is an involution. A direct computation shows that ||a||−1 = ||a′||. The following Lemma is used below.
LEMMA 1 Given vectors a and b if a + b ≠ 0, then a′ + b′ ≠ 0.
Proof Suppose a′ + b′ = 0. Then a/||a||² + b/||b||² = 0. Since a′ = −b′, we have ||a′|| = ||b′|| or ||a|| = ||b||, which implies a + b = 0. ¤
The involution considered above may be viewed as the “inverse” of a vector, and since Anderson and Duffin used the inverse of a matrix to define the parallel [reciprocal] sum of matrices, we define the reciprocal [parallel] sum of vectors a and b as follows: a reciprocally added to b, denoted by a ⊞ b, is given by
a ⊞ b = (a′ + b′)′.
From Lemma 1, we see that the definition is appropriate when a + b ≠ 0. In the case that a = 0 or b = 0, we define a ⊞ b = 0.
The following theorem summarizes some basic properties of the vector operation parallel addition; the proof consists of direct calculations and is omitted.
THEOREM 2 Given vectors a and b with a + b ≠ 0, then
(i) a ⊞ b = b ⊞ a, [commutative]
(ii) a ⊞ a = a/2, [non-reflexive]
(iii) (ka) ⊞ (kb) = k(a ⊞ b). [distributive]
A deeper result is that the reciprocal sum of vectors is an associative operation.
THEOREM 3 Given vectors a, b, and c, if a ⊞ (b ⊞ c) and (a ⊞ b) ⊞ c are both defined, then they are equal. And they both equal (a′ + b′ + c′)′.
THEOREM 5 Given vectors a and b with a + b ≠ 0, the reciprocal sum a ⊞ b may be written as follows:
a ⊞ b = (a||b||² + b||a||²) / (||a + b||²).
… The triangle inequality implies that
||a ⊞ b|| ≥ ||a|| ||b|| / (||a|| + ||b||),
and since k ⊞ m = km/(k + m) for scalars k and m, we have shown the following result.
COROLLARY 6 ||a ⊞ b|| = ||a|| ||b|| / ||a + b||, and ||a ⊞ b|| ≥ ||a|| ⊞ ||b||.