Some Properties for Linear Transformations on Self-Dual and Symmetric Cones
|School||Harbin Institute of Technology|
|Keywords||Euclidean Jordan algebra E- property Lyapunov transformations linear complementary problems automorphism invariance|
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 P-matrix 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 P-property, Q-property, Jordan P-property, order P-property 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 P-property, Q-property 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 0-property and order P-property.Finally, let (V , ) be an Euclidean Jordan algebra and K is its cone of squares. We consider a specific linear transformation--Lyapunov transformation La and give some necessary and sufficient conditions of E- property, Q- property, R 0-property 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 Q-property.(2) If the linear transformation L has E- property, then it has Q-property.(3) If the linear transformation L has positive principal minor property, then it has R 0- property.(4) Order P-property and E 0-property are algebra automorphism invariant in simple Euclidean Jordan algebras.(5) E-property is algebra automorphism invariant in any Euclidean Jordan algebra.(6) La has the Q-property La has the positive principal minor property a∈int( K).(7) La has the R 0-property 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.