A Matrix Inequality for the Inversions of the Restrictions 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.