Two calibration constants of a detector have measured standard deviations mm and mm

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

Two calibration constants of a detector have measured standard deviations

\sigma_1 = 4

mm and

\sigma_2 = 5

mm, with covariance matrix

\mathsf{C} = \begin{pmatrix}\sigma_1^2 & \rho\,\sigma_1\sigma_2\\ \rho\,\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}.

((a)) For a correlation coefficient

\rho = 0.6

, construct the upper-triangular Cholesky factor

\mathsf{R}

with

\mathsf{C} = \mathsf{R}^{\rm T}\mathsf{R}

, giving

r_{11}

r_{12}

and

r_{22}

. ((b)) Repeat the calculation for

\rho = 1.3

and show that the factorisation fails; give the value of the pivot at which it fails and explain what this means physically.

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