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.