A. S. Lewis

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A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X, 1 (X) 2 (X) : : : n (X), and hence may be written f(1 (X); 2 (X); : : :; n (X)) for some symmetric function f. Such functions appear in a wide variety of matrix optimization problems. We give a simple proof that this spectral function is diierentiable at X(More)
We characterize the exposed faces of convex sets C of symmetric matrices, invariant under orthogonal similarity (U T CU = C for all orthogonal U). Such sets C are exactly those determined by eigenvalue constraints: typical examples are the positive semideenite cone, and unit balls of common matrix norms. The set D of all diagonal matrices in C is known to(More)
We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby(More)
Synthetic aperture side-scan sonar (SAS) is an imaging modality for detecting objects on the sea floor and in shallow water. SAS images provide an echo of an object along with its acoustic shadow; both of which can be used in the classification and localization of the object. We developed a Fuzzy C-Means (FCM) image segmentation algorithm that segments the(More)
I outline a uniied approach to characterizing Fr echet, limiting Fr echet, and Clarke subgradients of an arbitrary function of the eigen-values of a real symmetric matrix. In particular, I compute various subdiierentials of the k'th largest eigenvalue. This paper summarizes the results and techniques presented in detail in 4]. 1 The author wishes to thank(More)
Dentofacial orthopaedics in relation to chronological age, growth period and skeletal development. An analysis of 72 male patients with Class II division 1 malocclusion treated with the Herbst appliance Urban Hiigg and Hans Pencberz 1 69 Mandibular rotations: concepts and terminology Beni Solow and William J. B. Houston 177 Marginal bone support and tooth(More)
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