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Moore–Penrose generalized inverse of the gradient tensor in Euler's equation for locating a magnetic dipole

### Abstract

Euler's equation provides us with a system of linear equations for localizing a magnetic dipole from measurements of the magnetic field and its gradients. However, so far, the condition for the coefficient matrix of the linear equations to be singular has not been shown. In this paper, we show that the matrix is singular if and only if the dipole moment is perpendicular to the dipole position vector, where the observation point is set at the origin. Moreover, we show that, even in this case, the true position can be uniquely reconstructed by using the Moore–Penrose generalized inverse of the gradient tensor.

© 2014 AIP Publishing LLC

Received 19 September 2013
Accepted 14 October 2013
Published online 17 January 2014

Acknowledgments:
This work was partially supported by the Moritani Scholarship Foundation, the Telecommunications Advancement Foundation, and Grant-in-Aid for Scientific Research (C) No. 24560072.

Article outline:

I. INTRODUCTION
II. THEORY
A. Euler's equation for localization of a magnetic dipole
B. Singularity of the gradient tensor and unique reconstruction using the Moore–Penroze generalized inverse
III. PLANAR SENSING
IV. EXPERIMENTS
V. CONCLUSION