TWO DIRECTIONS- with matrix unknowns
!. Matrix inequalities in matrix unknowns:
Extending Real Algebraic Geometry
subject of this talk
II. LMIs and Convex problems with matrix unknowns not much in this
talk
A systematic approach to operator and matrix inequalities:
noncommutative real algebraic geometry
Noncommutative Real Algebraic Geometry: Inequalities
Free Convexity
Free Convex Positvstellensatz
Noncommutative Real Nullstellensatz
Classical Real NullSS(Commutative)
Noncommutative Radicals
Noncommutative Real Nullstellensatz
Monomial ideals and unshrinkable words
A Positivstellensatz for Varieties Intersect a Convex Set
nc Polynomials
Definition
A noncommutative, or nc, polynomial p(x) is a polynomial in free
variables with involution x = (x 1 , . . . , x g , x ∗1 , . . . , x ∗g ).
Example p(x) = x 21 x ∗2 x 1 + x ∗1 x 2 x 21 + x 1 x 2 − x 2 (x 21 )∗ + 5.
Denote the nc polynomials with real coefficients by Rhx, x ∗ i.
Evaluating nc Polynomials
Given a g -tuple (X 1 , . . . , X g ) of real n × n matrices, p(X ) is defined
by plugging in matrices X j for each x j in p (and X ∗i for x ∗i ), and
plugging in p(0)In for the constant term p(0).
Example x j 7→ X j
x ∗j 7→ X ∗j
p(x) = 3 + x 21 + x ∗2 x 2
=⇒
p(X ) = 3In + X 21 + X ∗2 X 2 .
In the begining of this talk the X j are symmetric matrices and x ∗j = x j .
Noncommutative inequalities
Definition
(1) M 0 means M is positive semidefinite (PsD).
(2) M B means M − B 0.
(3) For r an nc polynomial let
Dr := {X = (X 1 , X 2 , . . . , X g ) : r (X ) 0}.
Here X j sweep thru symmetric matrices of all sizes.
Questions
Given q, r nc polynomials. How do we tell:
(1) Is q PsD everywhere?
(2) Is q(X ) PsD
if r (X ) is PsD? Equivalently, is q 0 on Dr ?
FIND ALGEBRAIC CERTIFICATES
Linear Pencil, Linear Matrix Inequality (LMI)
A free linear matrix inequality (LMI) is of the form L(X ) 0.
Its scalar solution set
DL = X ∈ Rs | L(X ) 0
= X ∈ Rs | A0 + X 1 A1 + · · · + X g Ag 0
is called a spectrahedron or also an LMI domain.
Theorem
H-McCullough Annals 2012
IF p is a nc symmetric polynomial p(0) = 1 with Dp bounded,
THEN there is a monic linear pencil L such that DL = Dp .
Linear Pencils
Given a s × s linear pencil
L(x) = A0 + x1 A1 + · · · + xg Ag ,
it is natural to substitute symmetric matrices X j for the variables xj .
For X = (X 1 , . . . , X g ) ∈ (Sn×n )g , the evaluation L(X ) is
L(X ) := In ⊗ A0 + X 1 ⊗ A1 + · · · + X g ⊗ Ag ∈ Ssn×sn .
The tensor product in this expression is the standard (Kronecker)
tensor product of matrices.
L is called monic if A0 = I .
Convex (perfect) Positivstellensatz
Theorem
H Klep McCullough Advances 2012
Suppose:
L(x) is a monic linear pencil;
q(x) is a noncommutative polynomial.
DL1 is bounded.
q 0 where L 0
if and only if
q(x) = s(x)∗ s(x) +
X
vj (x)∗ L(x)vj (x),
j
where s, vj are vectors of polynomials each of degree ≤
(If DL is bounded, then we may take s = 0.)
j
deg(q)
2
k
.
Noncommutative inequalities
Why care?
Applications to quantum physics are explained by Pironio, Navascués, Acı́n (SIAM J.
Optim. 20 (2010), pp. 2157–2180), who also consider computational aspects related
to nc sum of squares.
Optimization of nc polynomials has direct applications in quantum information
science, e.g. to compute upper bounds on the maximal violation of a generic Bell
inequality (Pal & Vértesi, Phys. Rev. A (3) 79 (2009), 022120).
Doherty, Liang, Toner, Wehner (Proceedings of the 23rd Annual IEEE Conference on
Computational Complexity, IEEE Computer Society, 2008, pp. 199–210) employ free
real algebraic geometry (or free positivity) to consider the quantum moment problem
and multi-prover games.
Optimization of nc polynomials is also used in quantum chemistry to compute the
ground-state electronic energy of atoms or molecules, cf. Mazziotti (Phys. Rev. Lett.
93 (2004), 213001).
Certificates of positivity via sums of squares are often used in the theoretical physics
literature to place very general bounds on quantum correlations, cf. Glauber (Phys.
Rev. 130 (1963), pp. 2529–2539).
nc Nullstellensatz
Chris Nelson
Jaka Cimprič
Igor Klep
Scott McCullough
UC San Diego
U Ljubljana
U Auckland
University of Florida
Your narrator Bill Helton
UCSD
AMS
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Try noncommutative computation
NCAlgebra1
NCSoStools2
http://www.math.ucsd.edu/~ncalg
http://ncsostools.fis.unm.si
Helton, de Oliveira (UCSD), Stankus (CalPoly SanLObispo), Miller
Igor Klep
Free Nullstellensatz
Noncommutative Real Algebraic Geometry: Inequalities
Free Convexity
Free Convex Positvstellensatz
Noncommutative Real Nullstellensatz
Classical Real NullSS(Commutative)
Noncommutative Radicals
Noncommutative Real Nullstellensatz
Monomial ideals and unshrinkable words
A Positivstellensatz for Varieties Intersect a Convex Set
Classical Real Algebraic Geometry
Classical Commutative Nullstellensätze (NullSS)
Commutative polynomials on Cg or Rg .
Variety (zero set) of p: Given polynomial p, the zero set Vp of p is
Vp := {X : p(X ) = 0}
Theorem (Hilbert’s NullSS) Polynomials on Cg , Vp ⊆ Cg :
Vq ⊇ Vp iff q k = f p for some polynomial f , some integer k.
Theorem (Real NullSS)
Polynomials on Rg , Vp ⊆ Rg :
2k
Vq ⊇ Vp iff q + sos = f p for some polynomial f , some integer k.
sos means sum of squares of polynomials =
finite
X
sj2 .
j
Algebra formulas are much more computer friendly than brute force
checking Vq ⊇ Vp .
nc Polynomials
For the remainder of this talk the variables and matrices are not
restricted to be symmetric.
Evaluating nc Polynomials
Given a g -tuple (X 1 , . . . , X g ) of real n × n matrices, p(X ) is defined
by plugging in matrices X j for each x j in p (and X ∗i for x ∗i ), and
plugging in p(0)In for the constant term p(0).
Example
x j 7→ X j
x ∗j 7→ X ∗j
p(x) = 3 + x 21 + x ∗2 x 2
=⇒
p(X ) = 3In + X 21 + X ∗2 X 2 .
Ideal Question
The left ideal generated by p is:
Ip := {f p : f a polynomial}
Is Ip all polynomials which vanish where p vanishes?
Often Not!
Main Question: Given p. Characterize
p
Ip := {q : p(X )v = 0 implies q(X )v = 0}
p
Ip ⊆ Ip called the radical ideal of Ip .
We want an Algebraic Certificate telling which q vanish where p
vanishes.
Example: “Analytic” Ideals
Definition
A polynomial p ∈ Rhx, x ∗ i is analytic if it is in Rhxi—that is,
p has no transpose variables, no x ∗i .
Example
Analytic:
Not analytic:
p(x) = 1 + x 1 x 2 + x 31 + x 52 .
p(x) = 1 + x ∗1 x 2 + (x ∗1 )3 + x 52 .
Theorem (H - McCullough - Putinar, 2007)
Let p ∈ Rhx, x ∗ i be an analytic polynomial, and let Ip ⊆ Rhx, x ∗ i be
the left ideal generated by p. Then
p
Ip = Ip ,
that is,
p(X )v = 0 =⇒ q(X )v = 0
iff
q = f p.
Note: no powers q k are needed! (Hilbert’s NullSS needs them.)
Classical Commutative Real NullSS
Commutative real valued polynomials on Rg or Cg .
Theorem (Real NullSS 1960’s): p and q are real valued polynomials
p
Ip
=
{q : q 2k + sos ∈ Ip } =:
√
real
I
The radical ideal of Ip equals the real radical ideal of Ip .
Algebra formulas are more computer friendly than brute force checking
of zero set inclusion.
Noncommutative Real Ideals
Definition
A left ideal I ⊆ Rhx, x ∗ i is called a real left ideal provided:
q1∗ q1 + sos ∈ I + I ∗
implies
q1 ∈ I .
∗
Commutative analog: q1k1 q1k1 + sos ∈ I implies q1 in I .
Proposition
If I ⊆ Rhx, x ∗ i is a left ideal, then
√
I is a real left ideal .
Noncommutative Real Radical Ideals
Definition
√
Given a left ideal I ⊂ Rhx, x ∗ i, define the real radical real I to be
√
real
I := the smallest real ideal containing I
So by the previous proposition,
I ⊆
¿ Free NullSS →
√
real
√
real
√
I ⊆ I
√
I = I?
← Free NullSS ?
Noncommutative Real Nullstellensatz
Theorem (nc NullSS Cimpric H McCullough Nelson, Pr LMSoc )
If I ⊆ Rhx, x ∗ i is a finitely-generated left ideal, then
√
√
real
I =
I.
Theorem
Let I ⊆ Rhx, x ∗ i be a finitely-generated left ideal whose generators
√
have degree ≤ d. There exists an ALGORITHM for computing real I :
1
2
This Algorithm involves only computations of polynomials which
have degree less than d. Stops in a finite number of steps.
√
The output is a left Gröbner Basis for the ideal real I .
Does I =
√
I?
We know
√
I =
√
real
I
Given generators of I can we tell ELEGANTLY by inspection if I =
√
I?
Solved for: degree 2 singly generated left ideal in one variable.
Answer is very messy (umpteen cases), SO we believe answer is NO.
All we can hope for is elegant cases. Example: Monomial Ideals
A monomial ideal is one whose generators are all monomials.
A monomial w ∈ hx, x ∗ i is left shrinkable if it can be written as
w = uu ∗ d for some monomials d, u with u 6= 1.
Example:
x2 x3 x ∗3 x ∗2 x1 is shrinkable.
Theorem Cimpric Klep Helton McCullough Nelson -arXiv
A left monomial ideal is a real left ideal if and only if it is generated by
left unshrinkable monomials.
Corollary
If generators have no x i x ∗i or x ∗i x i , then I is real.
Linear Pencils and Ideals
A POSITIVSTELLENSATZ ON A FREE VARIETY
INTERSECTED A FREE CONVEX SET
L-Radical Left Ideals
If I ⊂ Rhx, x ∗ i is a left ideal and L is a linear pencil, we say I is
L-radical if whenever
finite
X
i
si∗ si
+
finite
X
rj∗ Lrj ∈ I + I ∗
j
then each si ∈ I and each row of each Lrj is in I .
√
Let L I denote the smallest L-radical left ideal containing I and call it
the L-radical of I . Note that
√
√
real
L
I ⊂ I.
I ⊂
A Positivstellensatz
Theorem Chris Nelson in prep
Let I ⊂ Rhx, x ∗ i be a finitely-generated left ideal and let L be a linear
pencil. Then v ∗ p(X )v ≥ 0 whenever L(X ) 0 and ι(X )v = 0 for
each ι ∈ I if and only if p is of the form
q=
finite
X
i
where θ ∈
√
L
si∗ si +
finite
X
rj∗ Lrj + θ + θ∗ ,
j
I and each of the si , rj , and θ have degree bounds.
Nelson gave an ALGORITHM for finding all polynomials in
degree bounded by D, the degree of the generators of I .
√
L
I of
A Few Consequences’
Corollary Klep -Schweighofer 2011
If L is a linear pencil and L̃ is a degree 1 polynomial, then p(x) ≥ 0 for
all x ∈ DL if and only if L̃ is of the form
L̃ =
finite
X
i
qi∗ qi
+
finite
X
rj∗ Lrj + θ + θ∗ ,
j
where qi , ri are constants and there is a set special T of degree 1
polynomials containing the θi .
p
Nelson identifies the set T as generators for the left ideal L {0}. (It is
actually a two-sided ideal.) Also L can be any poly (KS, N).
Corollary Convex Perfect Positivstellensatz H- Klep McC
The Convex Positivstellensatz is the special case:
I = {0} and L is a monic pencil.
Summary: Free Nullstellensatz
√ √
Free real NullSS:
real
I ⊆
I = I
for finitely generated left ideals I .
√ √
Examples where I is a Real ideal, I = real I = I :
1
2
For an analytic finitely generated ideal.
√
One “sees” if I = real I
a. For a monomial ideal (unshrinkable words are bad)
b. Homogeneous singly generated ideals.
Nelson’s general NSS ∪ PosSS is much more general than Arveson
-Steinspring Thms (for finite dims).
ALGORITHM: It computes
√
real
Can Numerically Compute
I.
if q(X )v = 0 whenever p(X )v = 0 .
IMPLEMENTED under NCAlgebra by Igor Klep with M. de Oliveira.
Computer Algebra to do Noncommutative Polynomials
NCSoSTools Igor Klep et al. runs under Matlab.
NCAlgebra runs under Mathematica
M de Oliveira, Bill, Stankus, Miller (Igor helps too)
Now has a numerical LMI solver.
COMING ATTRACTIONS
Noncommutative Formal Power Series Commands
Lance Berlin, Luis A. Duffaut Espinosa, W. Steven Gray,
de Oliveira, Helton
References
J. William Helton, Igor Klep, Scott McCullough:
The convex Positivstellensatz in a free algebra,
Adv. Math. 231 (2012) 516–534.
Igor Klep, Markus Schweighofer:
Infeasibility certificates for linear matrix inequalities,
Oberwolfach Preprints (OWP) 28 (2011).
Jaka Cimprič, J. William Helton, Scott McCullough, Chris Nelson:
A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real
Ideal; Algorithms, preprint, http://arxiv.org/abs/1102.4859 (Pr LMS)
Jaka Cimprič, J. William Helton, Igor Klep, Scott McCullough, Chris Nelson:
On real one-sided ideals in a free algebra, in preparation.
Chris Nelson:
A Positivstellensatz on a Free Variety Intersected a Free Convex Set, in preparation.