iSoul In the beginning is reality.

# Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Harmonic algebra.

The vector inverse x−1 is defined as

$\mathbf{x}^{-1}=\frac{\mathbf{x}}{\|x\|^{2}}=\frac{\mathbf{x}}{\mathbf{x}\cdot&space;\mathbf{x}}$

with positive norm. For a non-zero scalar k,

$(k\mathbf{x})^{-1}=\frac{k\mathbf{x}}{\|k\mathbf{x}\|^{2}}=\frac{\mathbf{\mathbf{x}}}{k\|\mathbf{x}\|^{2}}=\frac{1}{k}\mathbf{x}^{-1}$

$\|\mathbf{x}^{-1}\|&space;=&space;\|\mathbf{x}||^{-1}$

The harmonic (or parallel) sum is usually symbolized by a colon (:), but I prefer a circle plus to maintain its relation with addition. The harmonic sum of vectors x and y is defined as

$\mathbf{x}\oplus&space;\mathbf{y}=\left&space;[&space;\mathbf{x}^{-1}+\mathbf{y}^{-1}&space;\right&space;]^{-1}$

if x0, y0, and x + y0; otherwise the sum equals 0.

The harmonic sum is commutative and associative, among other properties. Given two vectors with x + y0 and non-zero scalar k,

$\newline&space;\mathbf{x}&space;\oplus&space;\mathbf{y}&space;=&space;\mathbf{y}&space;\oplus&space;\mathbf{x}&space;\newline&space;\newline&space;\mathbf{x}&space;\oplus&space;(\mathbf{y}&space;\oplus&space;\mathbf{z})&space;=&space;(\mathbf{x}&space;\oplus&space;\mathbf{y})&space;\oplus&space;\mathbf{z}&space;\newline&space;\newline&space;\mathbf{x}&space;\oplus&space;\mathbf{x}&space;=&space;\mathbf{x}/2&space;\newline&space;\newline&space;(k\mathbf{x})&space;\oplus&space;(k\mathbf{y})=k(\mathbf{x}&space;\oplus&space;\mathbf{y})&space;\newline&space;\newline&space;\mathbf{x}&space;\oplus&space;\mathbf{y}&space;=&space;\frac{\mathbf{x}&space;\|\mathbf{y}\|^{2}+&space;\mathbf{y}\|\mathbf{x}\|^{2}}{\|\mathbf{x}+\mathbf{y}\|}$

The arithmetic mean for vectors x and y is

$A(\mathbf{x},\mathbf{y})&space;=&space;(\mathbf{x}+\mathbf{y})/2$

The harmonic mean for vectors x and y is

$H(\mathbf{x},&space;\mathbf{y})&space;=&space;2(\mathbf{x}&space;\oplus&space;\mathbf{y})$

The harmonic difference of vectors x and y such that x + y0 is defined as

$\mathbf{x}&space;\ominus&space;\mathbf{y}&space;=&space;\mathbf{x}&space;\oplus&space;(-\mathbf{y})$