How to calculate the variance in the solution to a set of linear equations?

A device measures counts on spectral channels, which are related to

the concentration of elements in a sample. Elements as in elements on the

pereodic table, sample as in a piece of material.

A measurement on the device results in a vector of spectral counts . These

counts are related to the composition of the sample measured .

is realated to using a second order tensor:

In order to find this relationship all elements of

must be found, thus, equations are needed to solve for all of .

Taking measurements on samples with known (and known distribution of

) results in equations per measurement. Using known samples

a set of linear equations is formed to solve for the coefficients .

contains the experimentally measured counts on the known reference

samples, contains the measured compositions (on other equipment) of

the reference samples, and is mapped onto a first order tensor.

By taking repeated measurements on the same physical samples, the variance of each member of and

is calculated. is solved for computationally by inverting and multiplying both sides, using Cramer's rule, row reducing or any number of computational methods. For example:

- Given that the variance of the members of and are known, how can

this information be used to compute the variance of the members of ? - When taking a measurement on a sample with unknown composition (once

has been calculated) how can the variance of be calculated?

Thanks for your help,

Mark