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 \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}\| = \|\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 \mathbf{y}=\left [ \mathbf{x}^{-1}+\mathbf{y}^{-1} \right ]^{-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 \mathbf{x} \oplus \mathbf{y} = \mathbf{y} \oplus \mathbf{x} \newline \newline \mathbf{x} \oplus (\mathbf{y} \oplus \mathbf{z}) = (\mathbf{x} \oplus \mathbf{y}) \oplus \mathbf{z} \newline \newline \mathbf{x} \oplus \mathbf{x} = \mathbf{x}/2 \newline \newline (k\mathbf{x}) \oplus (k\mathbf{y})=k(\mathbf{x} \oplus \mathbf{y}) \newline \newline \mathbf{x} \oplus \mathbf{y} = \frac{\mathbf{x} \|\mathbf{y}\|^{2}+ \mathbf{y}\|\mathbf{x}\|^{2}}{\|\mathbf{x}+\mathbf{y}\|}

The arithmetic mean for vectors x and y is

A(\mathbf{x},\mathbf{y}) = (\mathbf{x}+\mathbf{y})/2

The harmonic mean for vectors x and y is

H(\mathbf{x}, \mathbf{y}) = 2(\mathbf{x} \oplus \mathbf{y})

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

\mathbf{x} \ominus \mathbf{y} = \mathbf{x} \oplus (-\mathbf{y})

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