This preview shows page 1. Sign up to view the full content.
Unformatted text preview: n (of arbitrary dimension). In this exercise we prove several classical
characterizations of Euclidean distance matrices, derived by I. Schoenberg in the 1930s.
(a) Show that X is a Euclidean distance matrix if and only if
X = diag(Y )1T + 1 diag(Y )T − 2Y (23) for some matrix Y ∈ Sn (the symmetric positive semideﬁnite matrices of order n). Here,
diag(Y ) is the n-vector formed from the diagonal elements of Y , and 1 is the n-vector with
all its elements equal to one. The equality (23) is therefore equivalent to
xij = yii + yjj − 2yij , i, j = 1, . . . , n. Hint. Y is the Gram matrix associated with the vectors p1 , . . . , pn , i.e., the matrix with
elements yij = pT pj .
59 (b) Show that the set of Euclidean distance matrices is a convex cone.
(c) Show that X is a Euclidean distance matrix if and only if
X22 − X21 1T − 1X21 diag(X ) = 0, 0. (24) The subscripts refer to the partitioning
X21 X22 X= with X21 ∈ Rn−1 , and X22 ∈ Sn−1 .
Hint. The deﬁnition of Euclidean distance matrix involves only the dist...
View Full Document
This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
- Fall '13
- The Aeneid