Reciprocal sum of vectors

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, a1/2.

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 + b0, 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 ab, is given by

ab = (a′ + b′).

From Lemma 1, we see that the definition is appropriate when a + b0. In the case that a = 0 or b = 0, we define ab = 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 + b0, then

(i) ab = ba, [commutative]
(ii) aa = a/2, [non-reflexive]
(iii) (ka) ⊞ (kb) = k(ab). [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 ⊞ (bc) and (ab) ⊞ c are both defined, then they are equal. And they both equal (a′ + b′ + c′).

THEOREM 5     Given vectors a and b with a + b0, the reciprocal sum ab may be written as follows:

ab = (a||b||² + b||a||²) / (||a + b||²).

… The triangle inequality implies that

||ab|| ≥ ||a|| ||b|| / (||a|| + ||b||),

and since km = km/(k + m) for scalars k and m, we have shown the following result.

COROLLARY 6     ||ab|| = ||a|| ||b|| / ||a + b||, and ||ab|| ≥ ||a|| ⊞ ||b||.