A.S. LEWIS. Group invariance and convex matrix analysis. SIAM J. Matrix Anal. Appl., 17(4):927-- 949, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... n (x) t where i (x) is the ith largest eigenvalue of X. For any Elementary Algorithm for a Conic Linear System 8 p 2 [1; 1) let the norm of x be de ned by kxk = kxk p = 0 n X j=1 j j (x)j p 1 A 1 p ; i.e. kxk p is the L p norm of the vector of eigenvalues of x. See Lewis [14] for a proof that kxk p is a norm. When p = 1, kxk 1 is the sum of the absolute values of the eigenvalues of x. Therefore, when x 2 CX , kxk 1 = tr(x) n P i=1 x ii where x ij is the ijth entry of the real matrix x, and so kxk 1 is a linear function on CX . Therefore, when p = 1, we have ....

A.S. Lewis. Group invariance and convex matrix analysis. Technical Report, Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada, February 1995.

....triple of the Eaton triple (V, G, F ) if it is an Eaton triple and W : span F and H : g W : g # G, gW = W # O(W ) the orthogonal group of W . For x # V , let F (x) denote the unique element of the singleton set Gx # F . It is known that H is a finite reflection group [13] Also see [11, 16] for the normal decomposition systems and normal decomposition subsystems and their relation to Eaton triples and reduced triples. Let us recall some rudiments of finite reflection groups [8] Let V be a finite dimensional real inner product space. A reflection s # on V is an element of O(V ) ....

A.S. Lewis, Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl., 17:927949, 1996.

....of x. That is, x) 1 (x) n (x) t where i (x) is the i th largest eigenvalue of X . For any p 2 [1; 1) let the norm of x be defined by kxk = kxk p = 0 n X j=1 j j (x)j p 1 A 1 p ; i.e. kxk p is the L p norm of the vector of eigenvalues of x. see Lewis [14] for a proof that kxk p is a norm. When p = 1, kxk 1 is the sum of the absolute values of the eigenvalues of x. Therefore, when x 2 CX , kxk 1 = tr(x) n P i=1 x ii where x ij is the ijth entry of the Elementary Algorithm for a Conic Linear System 8 real matrix x, and so kxk 1 is a linear ....

A.S. Lewis. Group invariance and convex matrix analysis. Technical Report, Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada, February 1995.

....optimization, we do not pursue these refinements. 3 Group invariant convex analysis The more sophisticated convex analytic results outlined in the introduction arise elegantly by combining the framework of the Kostant Convexity Theorem with a straightforward algebraic structure introduced in [20]. Definition 3.1 Given a Euclidean space (E; h Delta; Deltai) a closed subgroup G of the orthogonal group O(E) and a G invariant map OE : E E, we say (E; G; OE) is a normal decomposition system if (i) for any point x in E there is a transformation in G satisfying (x) OE(x) and (ii) ....

....if and only if fl(y) 2 g(fl(x) holds and there exists an element k of the group K such that x and y both lie in (Ad k)D. Furthermore, g ffi fl is convex, or essentially strictly convex, or essentially smooth [24] if and only if g is likewise. Proof Apply Theorems 4.4, 4.5 and Corollary 6. 2 in [20]. We say a subset C of a Euclidean space is invariant under a group G of orthogonal linear transformations if C = C for any transformation in G. Theorem 3.6 (Invariant convex sets) An (AdK) invariant subset C of p is convex if and only if C a is convex. In this case an element x of p ....

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A.S. Lewis. Group invariance and convex matrix analysis. SIAM Journal on Matrix Analysis and Applications, 17:927--949, 1996.

....the eigenvalues of Z, arranged in decreasing order. This convexity theorem has a strong resemblance to a famous result of von Neumann [31] characterizing unitarily invariant matrix norms as symmetric gauge functions of the singular values. Indeed, the analogy is not accidental: the paper [19] develops an axiomatic framework subsuming both models, and at a more sophisticated level, both results follow quickly from the Kostant convexity theorem in semisimple Lie theory [20] The work we describe in this current paper also concerns the above type of convexity result, but with a very ....

A.S. LEWIS. Group invariance and convex matrix analysis. SIAM Journal on Matrix Analysis, 17(4):927--949, 1996.

....set consists of just one point, then that point is called exposed. The following corollary, although more straightforward to obtain directly (see [10] is a good illustration of the Exposed Faces Theorem. The analogous result for extreme points is also true, without the assumption of closure (see [11]) In particular this shows that if a subset C of R n is convex and permutation invariant then Diag C is a diagonal of Gamma1 (C) in the sense of [6] that is, a point in Diag C lies in ri (Diag C) if and only if it lies in ri ( Gamma1 (C) and is an extreme point of Diag C only if it ....

A.S. Lewis. Group invariance and convex matrix analysis. SIAM Journal on Matrix Analysis and Applications, 17:927--949, 1996.

....(5.14) The Subgradient Invariance Proposition now implies U:Diag y 2 (f ffi ) U:Diag x) or in other words U (x; y) 2 Graph (f ffi ) The arguments for the other subdifferentials are exactly analogous. 2 An eigenvalue function f ffi is convex if and only if the function f is convex (see [15]) In this case the regular subgradients are exactly the subgradients in the usual sense of convex analysis. The following result (c.f. 15] is easy to deduce from the regular case of the Subgradients Theorem. Corollary 6.12 (Convex Subgradients) Consider an eigenvalue function f ffi , where ....

....) The arguments for the other subdifferentials are exactly analogous. 2 An eigenvalue function f ffi is convex if and only if the function f is convex (see [15] In this case the regular subgradients are exactly the subgradients in the usual sense of convex analysis. The following result (c.f. [15]) is easy to deduce from the regular case of the Subgradients Theorem. Corollary 6.12 (Convex Subgradients) Consider an eigenvalue function f ffi , where the function f is convex. A matrix Y in S(n) is a (convex) subgradient of f ffi at a matrix X in S(n) if and only if X and Y have a ....

A.S. Lewis. Group invariance and convex matrix analysis. SIAM Journal on Matrix Analysis and Applications, 17, 1996. To appear.

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A.S. LEWIS. Group invariance and convex matrix analysis. SIAM J. Matrix Anal. Appl., 17(4):927-- 949, 1996.

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A.S. Lewis, Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl., 17:92720 949, 1996.

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