How far are two symmetric matrices from commuting? With an application to object characterisation and identification in metal detection

<p dir="ltr">Examining the extent to which measurements of rotation matrices are close to each other is challenging</p><p dir="ltr">due measurement noise. To overcome this, data is typically smoothed and Riemannian and Euclidean</p><p dir="ltr">metrics are applied. However, if rotation matrices are not directly measured and are instead formed by</p><p dir="ltr">eigenvectors of measured symmetric matrices, this can be problematic if the associated eigenvalues are</p><p dir="ltr">close. In this work, we propose novel semi-metrics that can be used to approximate the Riemannian metric</p><p dir="ltr">for small angles. Our new results do not require eigenvector information and are beneficial for measured</p><p dir="ltr">datasets. There are also issues when comparing rotational data arising from computational simulations</p><p dir="ltr">and it is important that the impact of the approximations on the computed outputs is properly assessed</p><p dir="ltr">to ensure that the approximations made and the finite precision arithmetic are not unduly polluting the</p><p dir="ltr">results. In this work, we examine data arising from object characterisation in metal detection using the</p><p dir="ltr">complex symmetric rank two magnetic polarizability tensor (MPT) description, we rigorously analyse</p><p dir="ltr">the effects of our numerical approximations and apply our new approximate measures of distance to the</p><p dir="ltr">commutator of the real and imaginary parts of the MPT to this application. Our new approximate measures</p><p dir="ltr">of distance provide additional feature information, which is invariant of the object orientation, to aid with</p><p dir="ltr">object identification using machine learning classifiers. We present Bayesian classification examples to</p><p dir="ltr">demonstrate the success of our approach.</p>