Changing Variables in Convex Matrix Ineqs

Transcription

1 Changing Variables in Convex Matrix Ineqs Scott McCullough University of Florida Nick Slinglend UCSD Victor Vinnikov Ben Gurion U of the Negev I am Bill Helton NCAlgebra a UCSD a Helton, Stankus (CalPoly SanLObispo ), Miller (Archemedies Inc )

2 NC CONVEX POLYNOMIALS 2

3 3 Function p of noncommutative variables x := (x 1, x 2 ) is MATRIX CONVEX (geometric def.) 0 α 1 p(α X + (1 α) Y ) αp( X) + (1 α)p( Y ) 1 2 p( X) p( Y ) p( 1 2 X + 1 Y )is Pos Def? 2 Question: Consider the noncommutative polynomial Is it matrix convex? p(x) := x 4 + (x 4 ) T.

4 CONVEX POLYNOMIALS ARE WIMPS 4 The riddle revealed: p(x) := x 4 + (x 4 ) T is not matrix convex. THM: ( McCullough + H) Every symmetric noncommutative polynomial which is matrix convex has degree 2 or less. COR: A Convex NC Poly is the Schur complement of some linear pencil. Every sublevel set of a convex polynomial is a ball.

5 NC CO-ORDINATE CHANGES 5 NONCOMMUTATIVE ANALYTIC CO-ORDINATE CHANGES Nick Slinglend - part of math thesis UCSD Variables x are g-tuples of free variables; no symmetry constraints on x s. Warm Up Exercise slichgvar

6 NC CO-ORDINATE CHANGES 6 An NC symm poly p, defines D 1 p := { X (R n n ) g : I p( X)}. An analytic polynomial map in x x f( x) := f 1 ( x). f k ( x) contains no x T j. f is a poly. f 1 likely a power series. Is there a bianalytic polynomial map f of D 1 p onto the unit ball D 1 (x T 1 x 1 + +x T g x g)? We restrict to hereditary defining polynomials, namely, p = j ht j g j where h j, g j are analytic.

7 BiAnalytic Equivalence 7 Is there a bianalytic map f of D p := { X : I p( X)}onto the unit ball D 1 (x T 1 x 1 + +x T g x g)? Commutative C case g = 1: solved by Riemann Map Thm. THEOREM N. Slinglend + H SUPPOSE that p( x) is a NC hereditary, matrix positive polynomial such that p(0) = 0. THEN We can construct a NC bianalytic map between D 1 p and D 1 (x T 1 x 1 + +x T g x g) using an LMI or no such maps exists. Surprising since construction is an LMI plus a rank constraint. Very nonconvex. We handle the rank constraint. a This gives an Algorithm

8 NC Analytic Maps of the Ball 8 NC ANALYTIC MAPS OF THE BALL

9 Ball Map Theorem 9 What are the NC analytic maps p of the unit ball D x T 1 x 1 + +x T g x g into the unit ball D x T 1 x 1 + +x T K x K which take the boundary into the boundary? K g We call these Ball Maps Commutative case De Angelo 1990ish. Simple ans if K < g 2 THEOREM N. Slinglend + H Every Ball Map, taking 0 to 0, is a linear isometry.

10 NC Plurisubharmonic Polynomials 10 NC PLURISUBHARMONIC POLYNOMIALS

11 NC Plurisubharmonic Polynomials 11 NOW: Polynomials which are not necessarily hereditary In (commutative) SCV the bianalytic transform (locally) of a convex function is plurisubharmonic. What does NC Plurisubharmonic (NC Plush) mean? For inspiration, what is a NC harmonic polynomial?

12 NC LAPLACIAN 12 NC Laplacian Lap[p, h] defined: For x 1, x 2 is d 2 dt 2[p((x 1 + th), x 2 )] t=0 + d2 dt 2[p(x 1, (x 2 + th))] t=0 Eg. Commuting Case: Lap[p, h] = h 2 x p(x) That is, Lap[p, h] = NCHess[p, {x 1, h}, {x 2, h}] The NC Laplacian is the formal trace of the NC Hessian. Harmonic NC polynomial p means Lap[p, h] = 0 for all x, h. Subharmonic NC polynomial p means Lap[p, H] is a pos semi def matrix for all X, H in SR n n.

13 MAIN THM (Holds Only in Two Vars)13 Define γ := x 1 + i x 2 here i 2 = 1 THM (J. Hernandez, McAllister, H -arxiv) (g = 2 only): For d > 2, the homogeneous NC symmetric polynomials in two symmetric variables which are (1) harmonic of degree d are precisely the linear combinations of Re(γ d ) and Im(γ d ) (2) subharmonic of degree 2d, are precisely the linear combinations: c 0 [Re(γ d )] 2 + c 1 Re(γ 2d ) + c 2 Im(γ 2d ) where c 0 0. Subharmonic of odd degree are harmonic.

14 BACK TO PLUSH 14 A symmetric NC polynomial p in free variables is p is called NC plurisubharmonic polynomial if its Levi operator is matrix positive. For p containing x and x T and we replace x T by y and define the following to be the NC Levi operator by L(p)[h] := 2 p t s (x + th, y + sk) t,s=0 y=x T,k=h T (1) Commuting Case: L(p)[h] := ij h i 2 p(x) x i x j h j

15 PLUSH POLYNOMIALS ARE: 15 L(p)[h] := 2 p t s (x + th, y + sk) t,s=0 y=x T,k=h T (2) Commuting Case: L((p))[h] = ij h i 2 p(x) x i x j h j Theorem 1. Vinnikov + H A symmetric NC polynomial p in free variables is NC plurisubharmonic iff p = f T j f j + g j g T j + F + F T (3) where each f j, g j, F is NC analytic.

16 Mixed Variables 16 NEW PART (Marching thru Slinglend s Thesis) MIXED VARIABLES x - free variables and y - symmetric variables p( x, y) - polynomial in mixed variables Engineering requires at least this level of generality.

17 Mixed Variables 17 x - free variables and y - symmetric variables p( x, y) - polynomial in mixed variables Mixed Analytic p is a polynomial in x j, y k but no x T j. Mixed Hereditary p has the form j ft j g j where f j, g j are mixed analytic. Eg p = x T 1 y 2x 1 + y 2 x T 1 y 1x 1 y 2. Given a Mixed Hereditary sym poly p(x, y). Is it Mixed-bianalytic to a Mixed Hered convex poly b(x, y)?

18 Theorem 18 A typical Mixed Hered convex poly b Not completely general b(x, y) = c+x T 1 x 1+ x T g free x gfree +y T 1 y 1+ y T g sym y gsym THM (Slinglend +H) SUPPOSE p is MixedHered sym poly.. THEN p = b R where b is convex poly and. R( x, y) is a bi -analytic poly IFF when p = p(0) + K j r j ( x, y) T r j ( x, y) r j polynomials THEN r( x, y) = Ω( y)r( x, y) where Ω( y) is a power series with Ω( Y ) an isometry on Range R( X, Y ), dim R is g free + g sym.

19 Suggests ALGORITHM FIND p = p(0) + K j r j( x, y) T r j ( x, y). THEN secretly we know r( x, y) = Ω( y)r( x, y) We want to find R( x, y) with dim R is g free + g sym? 2. CHECK FOR LINEAR DEPENDENCE of the r 1,, r K with coefficients which are functions only of y. If yes, then try to get unitarity. All details work fine when there is no y dependence.

20 THE END 20 THE END