ABSTRACT

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

1. 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 be analytic Hardy functions with index two. Then the following generalised isoperimetric inequality holds,

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 be the restriction of to an dimensional subspace of. Without loss of generality, assume that is the leading principal minor of. Let and denote the inverse of and of, respectively. Then we have the following results.

Theorem 1.1 If and is the vector of defined by, then

(1)

Here denotes conjugate transpose. As an immediate consequence, one also obtains the following corollary.

Corollary 1.2 If is the restriction of the matrix to, then

(2)

Here denotes the positive definite order, i.e., if and are square matrices, we say that if is a positive semi-definite matrix.

We observe that for, 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.

2. Proof of the Results

The proof of Theorem 1.1 is based on the theory of reproducing kernels. Therefore, we begin by introducing some notions and results which are used in the sequel.

2.1. 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 endowed with a function, which is called the reproducing kernel and which satisfies the reproducing property. Namely we have

(3)

and

(4)

for all and for all. We denote by (or) the reproducing kernel Hilbert space whose corresponding reproducing function is.

A complex-valued function is called a positive definite quadratic form function on the set, or shortly, positive definite function, if, for an arbitrary function and for any finite subset of, one has

(5)

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. So, in a sense, the correspondence between the reproducing kernel and the reproducing kernel Hilbert space is one to one.

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 be a strictly positive Hermitian matrix. Let

denote the inverse of. Then the space

of the complex valued functions on, endowed with the inner product

is a reproducing kernel Hilbert (complex Euclidean) space with reproducing kernel defined by for all.

Indeed, the validity of (3) follows by a straightforward calculation. To prove (4) we observe that

for all (note that). Thus coincides with the reproducing kernel Hilbert space. In particular the norm induced by the product coincides with the norm of.

We can thus combine the two theories of postitive definite Hermitian matrices and of reproducing kernels (see [4-12]).

2.2. 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 be a non-empty subset of. Let be a positive definite quadratic form function. Then the restriction of to is a positive definite quadratic form function on and the relation between

and is given by the following statement.

Proposition 2.1 (Restriction of RKHS) Let be a non-empty set and let be a non-empty subset of. Let be a positive definite quadratic form function. Then the Hilbert space defined by the positive definite quadratic form function is given by

(6)

Furthermore, the norm of is expressed in terms of the norm of by the following equality,

(7)

which holds for all.

See [1] for the details.

2.3. Proof of Theorem 1.1

Let with. Let and. Let be the positive definite quadratic form function on defined by for all Let and. Let be the function on defined by for all. Let. Then we have

and

. Thus (7)

implies that.

3. 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. Let be a positive definite Hermitian matrix and assume that

where is an matrix, is an matrix, and is an matrix. Observe that is positive definite and henceforth invertible. Then the inverse can be written in the form

where is the Schur complement with respect to. Since we also have which implies that. We now observe that the validity of Theorem 1.1 is equivalent to say that the matrix defined by

is positive semi-definite. Let and. Then we calculate

(here we understand that and are column vectors). Now we observe that

where denotes the Kronecker product of matrices. It is known that the Kronecker product of positive semidefinite matrices is positive semi-definite. Now

and, hence

is positive semi-definite and accordingly . Our proof is completed.

4. 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 realization 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.

5. 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.

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