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On the P-property of Z and
Lyapunov-like transformations on
Euclidean Jordan algebras
M. Seetharama Gowda
Department of Mathematics and Statistics
University of Maryland, Baltimore County
Baltimore, Maryland
gowda@math.umbc.edu
***************
GTORA Conference, Chennai
January 5-7,2012
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 1/20This is joint work with J. Tao and G. Ravindran.
Based on forthcoming paper in Linear Algebra and Its
Applications:
On the P-property of Z and Lyapunov-like transformations
on Euclidean Jordan algebras.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 2/20Outline
• Motivation and a conjecture
• Euclidean Jordan algebras
• Z and Lypaunov-like transformations
• Validity of the conjecture for Lyapunov-like transformations
• A result for Z-transformations
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 3/20Motivation
Recall a result from complementarity problems:
n×n
The following are equivalent for M ∈R :
All principal minors ofM are positive.
x∗Mx≤ 0⇒x = 0.
n
LCP(M,q) has a unique solution for all q∈R .
n
LCP(M,q): Find x∈R such that
x≥ 0, Mx+q≥ 0, and hMx +q,xi = 0.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 4/20WhenM is a Z-matrix, i.e., when all off-diagonal
entries of M are non-positive, the
above statements are further equivalent to:
LCP(M,q) has a solution for all q.
There exists ad> 0 such thatMd> 0.
M is positive stable: Real part of any
eigenvalue of M is positive.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 5/20n
S - All n×n real symmetric matrices.
n n
S - All PSD matrices inS .
+
n
Notation: X 0 if X ∈S .
+
hX,Yi :=trace(XY).
XY+YX
X◦Y := - Jordan product.
2
Semidefinite LCP:
n n n
L :S →S linear, Q∈S .
n
SDLCP(L,Q): FindX ∈S such that
X 0, L(X)+Q 0, andhX,L(X) +Qi = 0.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 6/20n×n
ForA∈R ,
T n
L (X) :=AX +XA - Lyapunov transformation onS .
A
T n
S (X) :=X−AXA - Stein transformation onS .
A
L denotes eitherL orS .
A A
Gowda-Song (2000), Gowda-Parthasarathy (2000):
The following are equivalent:
[XL(X) =L(X)X, X◦L(X) 0]⇒X = 0.
SDLCP(L,Q) has a solution for all Q.
There existsD≻ 0 with L(D)≻ 0.
L is positive stable.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 7/20The above result is very similar to the matrix theory
result for Z-matrices.
Why is this happening?
DoL andS have some sort of Z-property?
A A
Can the two results be unified and extended?
n n
Note: BothR andS are Euclidean Jordan algebras!
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 8/20(V,h·,·i,◦) is a Euclidean Jordan algebra if
V is a finite dimensional real inner product space
and the bilinear Jordan productx◦y satisfies:
x◦y =y◦x
2 2
x◦(x ◦y) =x ◦(x◦y)
hx◦y,zi =hx,y◦zi
2
K ={x :x∈V} is the symmetric cone inV .
Notation: x≥ 0 if x∈K andx> 0 if x∈int(K).
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 9/20Any EJA is a product of the following:
n n×n
S = Herm(R ) - n×n real symmetric matrices.
n×n
Herm(C ) - n×n complex Hermitian matrices.
n×n
Herm(Q ) - n×n quaternion Hermitian matrices.
3×3
Herm(O ) - 3× 3 octonion Hermitian matrices.
n
L - Jordan spin algebra.
Fora∈V , L (x) :=a◦x.
a
a andb operator commute if L L =L L .
a a
b b
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 10/20
