Abstract
Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs – xy -convexity – is also considered. A characterization of xy -convex polynomials is given.
Highlights
Convexity and its matricial analogs arise naturally in many mathematical and engineering contexts
The dimensionfree or scalable matrix analog of convexity appears in many modern applications, such as linear systems engineering [BGFB94, SIG98], wireless communication [JB07], matrix means [And89, And94, Han81], perspective functions [Eff09, ENE11], random matrices and free probability [GS09] and noncommutative function theory [DK+, HMV06, HM04, DHM17, BM14]
Linear system problems specified by a signal flow diagram naturally give rise to matrix inequalities p(a, x) 0, where p is a polynomial, or more generally a rational function, in freely noncommuting variables
Summary
Convexity and its matricial analogs arise naturally in many mathematical and engineering contexts. The root butterfly realization, gives an algebraic certificate for partial convexity near points in the domain of r of the form (A, 0). We say a function f of two freely noncommuting variables is xy-convex on a free set D if f (V ∗(X, Y )V ) V ∗f (X, Y )V for all isometries V , and all X, Y ∈ D satisfying V ∗(XY )V = (V ∗XV )(V ∗Y V ). Such a pair ((X, Y ), V ) is called an xy-pair.
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