Abstract
This chapter focuses on two problems concerning the individual entries of an EDM. The first problem is how to determine a missing or an unknown entry of an EDM. We present two methods for solving this problem, the second of which yields a complete closed-form solution. The second problem is how far an entry of an EDM can deviate from its current value, assuming all other entries are kept unchanged, before the matrix stops being an EDM. We present explicit formulas for the intervals, within which, entries can vary, one at a time, if the matrix is to remain an EDM. Moreover, we present a characterization of those entries whose intervals have zero length; i.e., those entries where any deviation from their current values renders the matrix non-EDM.
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Notes
- 1.
Recall that a configuration of D is obtained by factorizing \(\mathcal{T}(D)\), the Gram matrix of D. Hence, all configuration matrices of D have a full column rank which is equal to the rank of \(\mathcal{T}(D)\). As a result, P and P′ are not configuration matrices of D in the technical sense. Nonetheless, \(D =\mathcal{ K}(PP^{T})\) and \(D =\mathcal{ K}(P'P'^{T})\).
References
A.Y. Alfakih, On yielding and jointly yielding entries of Euclidean distance matrices. Linear Algebra Appl. 556, 144–161 (2018)
M. Bakonyi, C.R. Johnson, The Euclidean distance matrix completion problem. SIAM J. Matrix Anal. Appl. 16, 646–654 (1995)
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Alfakih, A.Y. (2018). The Entries of EDMs. In: Euclidean Distance Matrices and Their Applications in Rigidity Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-97846-8_7
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DOI: https://doi.org/10.1007/978-3-319-97846-8_7
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