Abstract
The transformations to diagonalize potential energy matrices for coupled harmonic oscillators will also diagonalize the variance when written in matrix form. After a brief review of a geometrical interpretation of the variance, the transformations are described and an example is given.
Highlights
In this note, we will be interested in n independently chosen real numbers x,...,x with mean x x and in their variance which can be written as ()We begin by recalling a geometric interpretation for s2.Let the n-tuple (x,x2, ...,xn) represent a point P in n-dimensional Euclidean space (E(n))and let orthogonal unit vectors al,a2 a be defined in the positive x,xv...,x, directions, e=xa +xna respectively
After a bit of algebra, we see that the equation for the variance can be rewritten yet again as
The symmetries which simplified the potential energy function were derived from the rotations of the circle
Summary
Let orthogonal unit vectors al,a2 a be defined in the positive x,xv...,x,, directions, e=xa +xna respectively. S is proportional to the square of the distance from P to e. We first rewrite Equation 1 in matrix form: J.
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