Euclidean Distance Matrices and Their Applications in Rigidity Theory by Abdo Y. Alfakih

Author:Abdo Y. Alfakih
Language: eng
Format: epub, pdf
ISBN: 9783319978468
Publisher: Springer International Publishing

(6.8)

where P 3 is either vacuous or the zero matrix. As a result, if D is a nonspherical centrally symmetric EDM, then it is easy to verify that P T De = 0. Thus, the characteristic polynomial of D, in this case, reduces to

(6.9)

As a result, we have the following theorem.

Theorem 6.2 (Alfakih [4])

Let D be an n × n nonspherical centrally symmetric EDM with embedding dimension r and let B = −JDJ∕2. Then r of the negative eigenvalues of D are equal to the nonzero eigenvalues of (−2B), the (r + 1)th negative eigenvalue of D is equal to , and the Perron eigenvalue of D is equal to .

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