### 15.1 Minimum variance reference system

The minimum variance reference system, i.e., a reference system with one of its axes aligned
with a direction along whit the field has the smallest fluctuations (Sonnerup and Cahill, 1967).
This method provides information on the spatial distribution of the fluctuations of a given
vector.
Given a generic field B(x,y,z), the variance of its components is

Similarly, the variance of B along the direction would be given by

Let us assume, for sake of simplicity, that all the three components of fluctuate around zero,
then

Then, the variance can be written as

which can be written (omitting the sign of average ) as

This expression can be interpreted as a scalar product between a vector and another vector
whose components are the terms in parentheses. Moreover, these last ones can be expressed as a product
between a matrix built with the terms , , , , , and a vector
. Thus,

where

and

At this point, is a symmetric matrix and is the matrix of the quadratic form
which, in turn, is defined positive since it represents a variance. It is possible to determine a new
reference system such that the quadratic form does not contain mix terms,
i.e.,

Thus, the problem reduces to compute the eigenvalues and eigenvectors of the matrix .
The eigenvectors represent the axes of the new reference system, the eigenvalues indicate the variance along
these axes as shown in Figure 102.

At this point, since we know the components of unit vectors of the new reference system referred to the
old reference system, we can easily rotate any vector, defined in the old reference system, into the new
one.