A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix

We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.


Introduction and Results
By exploiting the general structure of reproducing kernel Hilbert spaces, it is possible to prove very interesting norm inequalities (see, e.g., [1,2]).A typical result is as follows.
Let be an N-ply connected regular domain whose boundary consists of disjoint analytic Jordan curves.Let Moreover, we can completely describe the cases for which we have the equality instead of the inequality here above.Without the theory of reproducing kernels, such a simple and beautiful inequality could not be derived (see [2,3] for the details).
In this paper we introduce a new inequality.Let be a positive definite Hermitian matrix.Let and let Here denotes conjugate transpose.As an immediate consequence, one also obtains the following corollary.
is the restriction of the matrix Here  denotes the positive definite order, i.e., if M and are square matrices, we say that  is a positive semi-definite matrix.We observe that for 2 n  , such results can be checked directly.However, for , the result of Theorem 1.1 is not intuitive and appears mysterious, at least at first glance.

Proof of the Results
The proof of Theorem 1.1 is based on the theory of W. X.MAI ET AL. 56 reproducing kernels.Therefore, we begin by introducing some notions and results which are used in the sequel.

Reproducing Kernels
Let be an arbitrary abstract (non-void) set.Let denote the set of all complex-valued functions on .A reproducing kernel Hilbert spaces (RKHS for short) on the set is a Hilbert space , which is called the reproducing kernel and which satisfies the repro- ducing property.Namely we have ( By a fundamental theorem, we know that, for any positive definite quadratic form function on , there exists a unique reproducing kernel Hilbert space on with reproducing kernel K E E K .So, in a sense, the correspondence between the reproducing kernel and the reproducing kernel Hilbert space A simple example of positive definite quadratic form function is a positive definite Hermitian matrix.
Example 2.1 Let be a set consisting of distinct points.Let of the complex valued functions on , endowed with the inner product is a reproducing kernel Hilbert (complex Euclidean) space with reproducing kernel defined by Indeed, the validity of (3) follows by a straightforward calculation.To prove (4) we observe that ).Thus

 
n A H E coincides with the reproducing kernel Hilbert space   K H E .In particular the norm induced by the product coincides with the norm of

Restriction of a Reproducing Kernel
The validity of Theorem 1.1 follows by the properties of the restriction of a reproducing kernel in a general setting.Let be a non-empty set and let 0 be a non-empty subset of . Let Furthermore, the norm of which holds for all See [1] for the details.

Proof of Theorem
. Thus (7) implies that

An Alternative Proof Based on Schur Complement
We provide in this section a direct proof of Theorem 1.1 based on the properties of the Schur complement (cf., e.g., [13]).Let with 0 A be a positive definite Hermitian matrix and assume that is an matrix, and is an A is positive definite and henceforth invertible.Then the inverse n A can be written in the form is the Schur complement with respect to m S A   A .Since we also have n which implies that .We now observe that the validity of Theorem 1.1 is equivalent to say that the matrix Let and , , where  denotes the Kronecker product of matrices.It is known that the Kronecker product of positive semidefinite matrices is positive semi-definite.Now 1 1 0 is positive semi-definite and accordingly .Our proof is completed.
    * , , x y M x y  0

Remark
The results in this paper were given implicitly in the extensive paper [14].However, such results were not explicitly stated in the corresponding Theorem (Ultimate re- alization of reproducing kernel Hilbert spaces).For this reason, we wrote this paper where we clearly present our Theorem 1.1.We note that such ideas have arisen to our attention while analysing the structure of the theorem from the viewpoint of the support vector machine for the practical calculation.

Acknowledgements
The last author wishes to express his sincere gratitude to Professor Tsuyoshi Ando for providing exciting informations on system theory and the Schur complement.He is supported in part by the Grant-in-Aid for the Scientific Research (C)(2) (No. 24540113).
The research of M. Dalla Riva was supported by FEDER funds through COMPETE-Operational Programme Factors of Competitiveness ("Programa Operacional Factores de Competitividade") and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), within project PEst-C/MAT/UI4106/2011 with the Compete number FCOMP-01-0124-FEDER-022690.The research was also supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia") with the research grant SFRH/ BPD/64437/2009 and by "Progetto di Ateneo: Singular perturbation problems for differential operators"-University of Padova.
be analytic Hardy functions with index two.Then the following generalised isoperimetric inequality holds,

Proposition 2 . 1 (
Restriction of RKHS) Let be a non-empty set and let 0 be a non-empty subset of .be a positive definite quadratic form function.Then the Hilbert space defined by the positive definite quadratic form function vectors).Now we observe that

definite quadratic form function on
the set , or shortly, positive definite function, if, for an arbitrary function and for any finite subset