A device measures counts on I spectral channels, which are related to
the concentration of J 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 c_i. These
counts are related to the composition of the sample measured r_j.
c_i is realated to r_j using a second order tensor:

<br />
    A_{ij}c_i = r_j<br />

In order to find this relationship all K=I\times J elements of
A_{ij} must be found, thus, K equations are needed to solve for all of A_{ij}.

Taking measurements on samples with known r_j (and known distribution of
r_j) results in J equations per measurement. Using I known samples
a set of linear equations is formed to solve for the coefficients A_{ij}.

C_{kl}A_k = R_k
0  < k \le K \nonumber
0  < l \le K \nonumber

C_{kl} contains the experimentally measured counts on the known reference
samples, R_k contains the measured compositions (on other equipment) of
the reference samples, and A_k is A_{ij} mapped onto a first order tensor.
By taking repeated measurements on the same physical samples, the variance of each member of C_{kl} and R_k
is calculated. A_k is solved for computationally by inverting C_{kl} and multiplying both sides, using Cramer's rule, row reducing or any number of computational methods. For example:

 A_k = C_{kl}^{-1} R_k

  1. Given that the variance of the members of C_{kl} and R_{k} are known, how can
    this information be used to compute the variance of the members of A_k?
  2. When taking a measurement on a sample with unknown composition (once
    A_{ij} has been calculated) how can the variance of r_j be calculated?


Thanks for your help,

Mark