Some Properties for Linear Transformations on SelfDual and Symmetric Cones 

Author  LiYuanMin 
Tutor  WangXingTao 
School  Harbin Institute of Technology 
Course  Basic mathematics 
Keywords  Euclidean Jordan algebra E property Lyapunov transformations linear complementary problems automorphism invariance 
CLC  O151.26 
Type  Master's thesis 
Year  2008 
Downloads  30 
Quotes  0 
It makes significant sense to study linear complementary problems (LCP) with the help of Euclidean Jordan algebraic technique. A real square matrix M is said to be a Pmatrix if all its principal minors are positive. It is well known that this property has various equivalent forms. Gowda extended these notions to a linear transformation defined on Euclidean Jordan algebras and he introduced Pproperty, Qproperty, Jordan Pproperty, order Pproperty and positive principal minor property etc. Also he studied their relationship. Based on his study, we introduce E 0 property and E property of linear transformations defined on Euclidean Jordan algebras and discuss their relation to Pproperty, Qproperty and positive principal minor property.In addition, Gowda introduced algebra automorphism invariance and cone automorphism invariance. Also he proved that the property related to Jordan product is algebra automorphism invariant and the property related to the solution of LCP is cone automorphism invariant. Based on his study, we make a further study mainly on the algebra automorphism invariance of E property, E 0property and order Pproperty.Finally, let (V , ) be an Euclidean Jordan algebra and K is its cone of squares. We consider a specific linear transformationLyapunov transformation La and give some necessary and sufficient conditions of E property, Q property, R 0property and positive principal minor property for La respectively. Also, we give a complementary form of the general Lyapunov theorem.The results we obtained are the followings(1) If the linear transformation L has E 0 property and R 0 property, then it has Qproperty.(2) If the linear transformation L has E property, then it has Qproperty.(3) If the linear transformation L has positive principal minor property, then it has R 0 property.(4) Order Pproperty and E 0property are algebra automorphism invariant in simple Euclidean Jordan algebras.(5) Eproperty is algebra automorphism invariant in any Euclidean Jordan algebra.(6) La has the Qproperty La has the positive principal minor property a∈int( K).(7) La has the R 0property if and only if a is invertible.(8)a∈int( K)if and only if for each q∈int( K), there exists x∈int( K) such that( )La x = q.