bv_cvxbook_extra_exercises

To nd f we can solve a feasibility problem nd ai bi

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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 . i 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 T X22 − X21 1T − 1X21 diag(X ) = 0, 0. (24) The subscripts refer to the partitioning T x11 X21 X21 X22 X= with X21 ∈ Rn−1 , and X22 ∈ Sn−1 . Hint. The definition of Euclidean distance matrix involves only the dist...
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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.

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