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*,

(*k***a**)**‘** = *k***a **/ ||k**a**||² = **a **/ *k*||**a**||² = **a′**/*k*.

Next we see that **a″** = (**a′**)′ = (**a**/||**a**||²)**′** = ||**a**||²**a′** = **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) (*k***a**) ⊞ (*k***b**) = *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**||.