In the setting of semidefinite linear complementarity problems on
S^{n}
, we focus on the Stein Transformation
S_{A}
(
X
) :=
X
−
AXA^{T}
, and show that
S_{A}
is (strictly) monotone if and only if
v_{r}
(
UAU^{T}
◦
UAU^{T}
) (<) ≤ 1, for all orthogonal matrices
U
where ◦ is the Hadamard product and
v_{r}
is the real numerical radius. In particular, we show that if
ρ
(A) < 1 and
v_{r}
(
UAU^{T}
◦
UAU^{T}
) ≤ 1, then SDLCP(
S_{A},Q
) has a unique solution for all
Q
∈
S^{n}
. In an attempt to characterize the
GUS
property of a nonmonotone
S_{A}
, we give an instance of a nonnormal 2 × 2 matrix A such that SDLCP(
S_{A},Q
) has a unique solution for
Q
either a diagonal or a symmetric positive or negative semidefinite matrix. We show that this particular
S_{A}
has the
P'_{2}
property.
AMS Mathematics Subject Classification : 90C33, 93D05.
1. Introduction
Given a continuous function
f
from a real Hilbert space
H
into itself and a closed convex set
K
in
H
, the
variational inequality problem
VI(
f,K
) is to find a vector
x
^{∗}
in
H
such that
This problem has been extensively studied in the literature. In the infinite dimensional setting, it appears in the study of partial differential equations, mechanics, etc.
[17]
. In the finite dimensional setting, it appears in optimization, economics, traffic equilibrium problems etc.
[13]
.
Now suppose
K
is a cone, i.e.,
tK
⊆
K
for all
t
≥ 0. Then by putting
x
= 0 and
x
= 2
x
^{∗}
, the condition
leads to
If we define the dual cone
K
^{∗}
of
K
by
then, (2) and (3) together imply that
f
(
x
^{∗}
) ∈
K
^{∗}
.
So when
K
is a closed convex cone, (1) becomes the problem of finding an
x
^{∗}
∈
H
such that
This is a
cone complementarity problem
. This problem and its special cases often arise in optimization (KarushKuhnTucker conditions), game theory (bimatrix games), mechanics (contact problem, structural engineering), economics (equilibrium in a competitive economy) etc. For a detailed description of these applications, we refer to
[16]
,
[6]
,
[5]
,
[20]
,
[21]
. In this paper, we focus on the socalled
semidefinite linear complementarity problem
(SDLCP) introduced by Gowda and Song
[8]
: Let
S
^{n}
denote the space of all real symmetric
n
×
n
matrices, and
be the set of symmetric positive semidefinite matrices in
S
^{n}
. With the inner product defined by ⟨
Z,W
⟩ :=
tr
(
Z W
), ∀
Z,W
∈
S
^{n}
, the space
S
^{n}
becomes a Hilbert space. Clearly,
is a closed convex cone in
S
^{n}
. Given a linear transformation
L
:
S
^{n}
→
S
^{n}
and a matrix
Q
∈
S
^{n}
, the
semidefinite linear complementarity problem
, denoted by SDLCP(
L,Q
), is the problem of finding a matrix
X
∈
S
^{n}
such that
Examples of the semidefinite linear complementarity problem are: the standard linear complementarity problem
[4]
, the block SDLCP
[27]
, and the geometric SDLCP of Kojima, Shindoh, and Hara
[18]
. For detalils on how to reformulate these as the SDLCP of Gowda and Song (6), we refer to the Ph.D. Thesis of Song (Section 1.3
[22]
). We give the description of the standard linear complementarity problem here that is needed in the paper. Consider the Euclidean space
R^{n}
with the cone of nonnegative vectors
Given a matrix
M
∈
R
^{n×n}
and a vector
q
∈
R^{n}
, the
linear complementarity problem
LCP(
M, q
) is to find a vector in
R^{n}
such that
where ⟨
x, y
⟩ is the usual inner product in
R^{n}
. This problem has been studied in great detail, see
[4]
,
[6]
,
[5]
,
[20]
. In this setting, we have the following result:

(a) (3.3.1, 3.3.7.[4]) LCP(M, q) has a unique solution for any givenq∈Rnif and only ifMis aPmatrix (that is, all its principal minors are positive).

(b) (3.3.4.[4])Mis aPmatrix if and only if
where
y_{i}
denotes the
i
th element of the vector
y
.
As mentioned above, the standard linear complementarity problem is a special case of the semidefinite linear complementarity problem in the following way
[22]
:
Let a matrix
M
∈
R
^{n×n}
and a vector
q
∈
R^{n}
be given. Define a linear transformation
M
:
S
^{n}
→
S
^{n}
by
M
(
X
) :=
Diag
(
Mdiag
(
X
)), where
diag
(
X
) is a vector whose entries are the diagonal entries of the matrix
X
, and
Diag
(
u
) is a diagonal matrix whose diagonal is the vector
u
. Corresponding to LCP(
M, q
) in (7), one can consider SDLCP(
M
,
Diag
(
q
)), which is to find
X
∈
S
^{n}
such that
If
X
is a solution of SDLCP(
M
,
Diag
(
q
)), then
diag
(
X
) solves LCP(
M, q
). Conversely, if
x
solves LCP(
M, q
), then
Diag
(
x
) solves SDLCP(
M
,
Diag
(
q
)). In this sense, these two complementarity problems are equivalent.
We want to note that the cone of nonnegative vectors
in
R^{n}
is polyhedral. That is,
is the intersection of a finite number of sets defined by linear inequalities. However, the cone of symmetric positive semidefinite matrices
in
S
^{n}
is not polyhedral. We want to note that because of the nonpolyhedrality of the cone
and the noncommutativity of the matrix product, the results available for the standard linear complementarity problem do not simply carry over to the SDLCPs. To ellaborate on this, for example, consider the socalled the
P
and
GUS
properties introduced by Gowda and Song
[8]
: A linear transformation
L
:
S
^{n}
→
S
^{n}
has the

(c)Pproperty if [XL(X) =L(X)Xnegative semidefinite] ⇒X= 0,

(d)Globally Uniquely Solvable (GUS)property if for allQ∈Sn, the semidefinite linear complementarity problem (6) SDLCP(L,Q) has a unique solution.
Note that (c) is an analogus definition to (b) and in the LCP case,
P
⇔
GUS
(see (a)). Does the equivalence still hold for SDLCPs? Authors
[7]
,
[1]
,
[12]
specialized these properties to the Lyapunov transformation
L_{A}
(
X
) :=
AX
+
XA^{T}
, the Stein transformation
S_{A}
(
X
) :=
X
−
AXA^{T}
, the twosided multiplication transformation
M_{A}
(
X
) :=
AXA^{T}
, and studied interconnections with the (global) asymptotic stability of the continuous and discrete linear dynamical systems (see Introduction of
[12]
)
Gowda and Song showed that (Thm 5
[8]
) the Lyapunov transformation
L_{A}
(
X
) :=
AX
+
XA^{T}
has the
P
property if and only if
A
is positive stable (i.e., every eigenvalue of
A
has a positive real part) and hence the continuous system in (8) is globally asymptotically stable
[19]
; and has (Thm 9
[8]
) the
GUS
property if and only if
A
is positive stable and positive semidefinite. Therefore, the
P
and
GUS
properties are
not
equivalent in SDLCP setting.
Bhimasankaram et al.
[1]
showed that for the twosided multiplication transformation
M_{A}
(
X
) :=
AXA^{T}
,
GUS
and
P
are both equivalent to
A
being positive definite or negative definite. Zhang
[28]
, in particular, looked at a problem of solving the matrix equation
AXA^{T}
+
BY B^{T}
=
C
seeking a solution
X
⪰ 0 for given matrices
A,B
and
C
and provided necessary and sufficient conditions.
Gowda and Parthasarathy (Theorem 11, Remark 4
[7]
) showed that the Stein transformation
S_{A}
(
X
) :=
X
−
AXA^{T}
has the
P
property if and only if
ρ
(
A
) < 1 and hence the discrete system in (8) is globally asymptotically stable
[23]
. However, the characterization of the
GUS
property of the Stein transformation is still open. The known results so far are in 2003, Gowda, Song, and Ravindran (Thm 3
[11]
) showed that if
S_{A}
is strictly monotone (that is, ⟨
X
;
S_{A}
(
X
)⟩ > 0 for all 0 ≠
X
∈
S
^{n}
), then
S_{A}
has the
GUS
property; If
A
is normal, the converse also holds. Moreover, in 2013, J. Tao
[26]
examined conditions which gives the socalled
Cone

GUS
property (that is, SDLCP(
S_{A}
,
Q
) has a unique solution for all
.
Therefore, the aims of this paper are to give a characterization of
S_{A}
being strictly monotone in terms of the real numerical radius of
A
and hence providing a sufficient condition for the
GUS
property of
S_{A}
; and to examine a particular instance of
A
so that SDLCP(
S_{A}
,
Q
) has a unique solution for all
Q
diagonal or symmetric
negative
semidefinite.
The main results of the paper
are as follows: In section 2, we show that the Stein Transformation
S_{A}
is (strictly) monotone if and only if
v_{r}
(
UAU^{T}
◦
UAU^{T}
) (<) ≤ 1 for all
U
orthogonal, where ◦ denotes the Hadamard product and
v_{r}
denotes the real numerical radius defined in this paper (see Section 2). In particular, if
ρ
(
A
) < 1 and
v_{r}
(
UAU^{T}
◦
UAU^{T}
) ≤ 1 then SDLCP (
S_{A}
,
Q
) has a unique solution for all
Q
∈
S
^{n}
. As a byproduct, we get a result that if
tr
(
AA^{T}
) >
n
, then
S_{A}
is not monotone. In section 3, we look at a particular case of the 2 × 2 matrix
A
(motivated by
[10]
):
and show that SDLCP(
S_{A},Q
) has a unique solution if
Q
is either a diagonal or a symmetric
negative
semidefinite matrix. Moreover, we show that not only
S_{A}
but also every principal subtransformations of
defined by
invertible (
X
∈
S
^{n}
) has the
Cone

Gus
properties.
We list below some necessary definitions:

(a) A linear transformationL:Sn→Snis calledmonotoneif ⟨L(X),X⟩ ≥ 0 ∀X∈Sn;strictly monotoneif ⟨L(X),X⟩ > 0 for all nonzeroX∈Sn

(b) A linear transformationL:Sn→Snis said to have the

ConeGuspropertyif for allQin the cone of symmetric positive semedefinite matrices, SDLCP(L,Q) has a unique solution.

P′2propertyifX⪰ 0;XL(X)X⪯ 0 ⇒X= 0.

Cross Commutative propertyif for anyQ∈Snand for any two solutionsX1andX2of SDLCP(L,Q),X1Y2=Y2X1andX2Y1=Y1X2, whereYi=L(Xi) +Q(i= 1; 2).

(c) A matrixM∈Rn×nis called

positive semidefiniteif ⟨Mx, x⟩ ≥ 0 for allx∈Rn. IfMis symmetric positive semidefinite, we use the notationM⪰ 0. The notationM⪯ 0 means −M⪰ 0. Note that a nonsymmetric matrixMis positive semidefinite if and only if the symmetric matrixM+MTis positive semidefinite.

positive definiteif ⟨Mx, x⟩ > 0 for all nonzerox∈Rn. IfMis symmetric positive definite, we use the notationM≻ 0.

Pmatrixif all its principal minors are positive.

orthogonalifMMT=MTM=I, whereIis then×nidentity matrix.

 normal ifMMT=MTM.
(d) The Hadamard product (or Schur product) of two matrices
A
and
B
is the entrywise product of
A
and
B
.
2. Characterization of the Monotonicity of the Stein Transformation
Recall
[14]
that the
Numerical Radius
of an
n
×
n
matrix
A
is defined as
v
(
A
) =
max
{
x^{T}Ax
 : ∥
x
∥ = 1,
x
∈
C^{n}
}, where ∥
x
∥ denotes the Euclidean norm of the vector
x
. Here we define the
real
version of the Numerical Radius for a real matrix
A
∈
R
^{n×n}
as
and relate it with the monotonicity of the Stein Transformation
S_{A}
.
Example
For
r
> 0, let
Since
x^{T}Ax
= 0 for all
x, v
(
A
) =
v_{r}
(
A
) = 0.
Note that for a square matrix
A
,
ρ
(
A
) ≤
v
(
A
) since each square matrix
A
has an eigenvector in
C^{n}
, but this inequality is not necessarily true for
v_{r}
(
A
). See in the above example that
and therefore
ρ
(
A
) =
r
>
v_{r}
(
A
), where
ρ
(
A
) denotes the spectral radius of
A
and
σ
(
A
) is the spectrum of
A
(i.e., the set of all eigenvalues of
A
). We also note that for a square matrix
A
,
because 
x^{T}Ax
 = 
x^{T}A^{T}x
 and
is a symmetric matrix.
We characterize the monotonicity of
S_{A}
in terms of the real numerical radius.
Theorem 2.1.
For A
∈
R
^{n×n}
,
the Stein Transformation S_{A}
:
S
^{n}
→
S
^{n}
,
S_{A}
(
X
) :=
X
−
AXA^{T}
is (strictly) monotone if and only if for all orthogonal matrices U
,
v_{r}
(
UAU^{T}
◦
UAU^{T}
) (<) ≤ 1.
Proof
. Suppose
S_{A}
is monotone. Then ⟨
S_{A}
(
D
);
D
⟩ ≥ 0, where
D
is a diagonal matrix with the diagonal equals a unit vector
d
. Note that
and hence
v_{r}
(
A
◦
A
) ≤ 1. Since
S_{A}
is monotone,
S_{UAUT}
is also monotone for all orthogonal matrices
U
. Therefore,
v_{r}
(
UAU^{T}
◦
UAU^{T}
) ≤ 1. For the converse, let
B
=
UAU^{T}
and suppose
v_{r}
(
B
◦
B
) ≤ 1 for all orthogonal matrices
U
. Take
X
∈
S
^{n}
, then
X
=
UDU^{T}
where
D
is a diagonal matrix with the diagonal
d
. Upon replacing
X
by
UDU^{T}
and carrying out the calculation we get
as
tr
(
BDB^{T}D
) = ⟨
d
, (
B
◦
B
)
d
⟩. By our assumption
v_{r}
(
B
◦
B
) ≤ 1, therefore ⟨
S_{A}
(
X
),
X
⟩ ≥ 0 for all
X
∈
S
^{n}
. Hence
S_{A}
is monotone.
The proof for strict monotonicty is similar

Remark 2.1.(a) If we letB=UAUT, thenvr(B◦B) can be computed asdenotes the operator norm induced by the Euclidean vector norm.

(b) Chen and Qi in 2006[3]introduced the CartesianPproperty which is equivalent to the strict monotonicty in some special case (p179[3]) and showed that (Corollary 1[3]) a linear transformationL:Sn→Snis strictly monotone if and only if for any 0 ≠X∈Snand anyUorthogonal, there exists an indexi∈ {1, · · · ,n} such that [UXL(X)UT]ii> 0, where [M]ijis the (i, j)component of the matrixM. We also gave sufficent and necessary conditions for monotocity in Theorem 2 above whenLis restricted toSA. We think our result is easier in view of computations involved because in Chen and Qi, it involves both random nonzero symmetric matricesXand orthogonal matricesUto verify the condition. However, our result involves only orthogonal matrices. There are numerical methods to radomly generate orthogonal matrices, for example, see[24], and the real numerical radiusvrcan again be computed numerically as the operator norm as mentioned in Part(a) above.
We now state a simple check condition for a
non
monotonicity of
S_{A}
below.
Corollary 2.2.
If tr
(
AA^{T}
) >
n for A
∈
R
^{n×n}
,
then S_{A} is not monotone.
Proof
. We shall show that if
S_{A}
is monotone, then
tr
(
AA^{T}
) ≤
n
.
First we observe that for a diagonal matrix
D
,
Therefore, if
S_{A}
is monotone, (
I
−
A
) ◦ (
I
+
A
) is positive semidefinite, as well as (
I
−
B
) ◦ (
I
+
B
) (for
S_{A}
monotone implies
S_{B}
monotone) for all
B
=
UAU^{T}
, where
U
is an arbitrary orthogonal matrix.
Suppose
S_{A}
is monotone. Then (
I
−
A
) ◦ (
I
+
A
) =
I
−
A
◦
A
is positive semidefinite which means ⟨
e
, (
I
−
A
◦
A
)
e
⟩ ≥ 0, where
e
is the vector of all 1’s. This reduces to ⟨
e
,
e
⟩−⟨
e
, (
A
◦
A
)
e
⟩ ≥ 0. Since ⟨
e
,
e
⟩ =
n
and ⟨
e
, (
A
◦
A
)
e
⟩ =
tr
(
AA^{T}
), we get the desired result.
We state below a sufficient condition for
S_{A}
to be
GUS
in terms of the matrix
A
.
Theorem 2.3.
For A
∈
R
^{n×n}
,
if p
(
A
) < 1,
v_{r}
(
UAU^{T}
◦
UAU^{T}
) ≤ 1
for all orthogonal matrices U, then S_{A} has the
GUS

property
.
Proof
. The condition
ρ
(
A
) < 1 is equivalent to
S_{A}
having the
P
property which is also equivalent to the existence of the solution to SDLCP(
S_{A}
,
Q
) for all
Q
∈
S
^{n}
. (Thm 11, Remark 4
[7]
). By Theorem 2,
S_{A}
is monotone. Suppose there are two solutions
X
_{1}
and
X
_{2}
to SDLCP (
S_{A}
,
Q
). Let
Y_{i}
=
S_{A}
(
X_{i}
) +
Q, i
= 1, 2. Thus, 0 ≤ ⟨
S_{A}
(
X
_{1}
−
X
_{2}
),
X
_{1}
−
X
_{2}
⟩ = ⟨
Y
_{1}
−
Y
_{2}
,
X
_{1}
−
X
_{2}
⟩ ≤ 0 since
tr
(
X
_{1}
Y
_{1}
) = 0 =
tr
(
X
_{2}
Y
_{2}
) and
X_{i}
,
Y_{i}
⪰ 0 for
i
= 1, 2. This leads to
tr
(
X
_{2}
Y
_{1}
) +
tr
(
X
_{1}
Y
_{2}
) = 0. Since the involved matrices are all positive semidefinite, each trace is nonnegative. Hence
tr
(
X
_{2}
X
_{1}
) = 0 =
tr
(
X
_{1}
Y
_{2}
), resulting
X
_{2}
Y
_{1}
= 0 =
X
_{1}
Y
_{2}
. Hence
S_{A}
has the
Cross Commutative
property. It is known (Thm 7
[8]
) that
P
+
CrossCommutativity
⇔
GUS
, hence the proof is complete.
Remark 2.2.
As it is seen in the above proof, monotonicity implies the Cross Commutativeproperty. Whether monotonicity is equivalent to the Cross Commutativeproperty is an open problem. This would complete the characterization of the
GUS
property of
S_{A}
.
3. On a special SA: S2→ S2
In what follows, let
R_{C}
(
X
) :=
Diag
(
Cx_{d}
)+
X
_{0}
, where
C
∈
R
^{n×n}
is a matrix,
x_{d}
is the vector composed of the main diagonal of the matrix
X
, and
X
_{0}
is the matrix obtained after replacing all the diagonal elements of
X
with zeros. If we let
and
C
=
I
−
A
◦
A
, then
Note that this
A
is not normal. Motivated by the question raised in (p12, Problem 2.
[10]
), we studied this particular instance of
S_{A}
. So far, the following is known:

(a)SAhas theGUSproperty whenϒ2≤ 2 (p12[10]).

(b)SAisNot GUSwhenϒ2> 4 (p12[10]).

(c)SAhas theConeGusproperty whenϒ2≤ 4. This is obtained by applying J. Tao’s result on this particularSA. In Corollary 4.1[25], it states thatSA:S2→S2has the ConeGus property if and only ifρ(A) < 1 andI±Aare positive semidefinite matrices.
In
[10]
, Gowda and Song raised the question:
We were able to show that this
S_{A}
has the
GUS
property regardless of the value of
ϒ
if the given
Q
is either diagonal or symmetric
negative
semidefinite. We start with a lemma which shows that every positive definite solution of an SDLCP(
L,Q
) is locally unique if
L
has the
P
property.
Lemma 3.1.
Suppose a linear transformation L
:
S
^{n}
→
S
^{n}
has the
P

property. Then SDLCP
(
L,Q)
cannot have two distinct positive definite solutions.
Proof
. Suppose
X
_{1}
and
X
_{2}
are two distinct positive definite solutions to a given SDLCP(
L,Q
). Then we get
X
_{1}
(
L
(
X
_{1}
) +
Q
) = 0 =
X
_{2}
(
L
(
X
_{2}
) +
Q
): Since both
X
_{1}
and
X
_{2}
are invertible, this would mean
L
(
X
_{1}
) = −
Q
=
L
(
X
_{2}
). Hence (
X
_{1}
−
X
_{2}
)
L
(
X
_{1}
−
X
_{2}
) = 0. By the
P
property of
L
,
X
_{1}
=
X
_{2}
contradicting our assumption.
Now, let’s consider our special
S_{A}
given in (9). Since all eigenvalues are zeros for any
ϒ
, note that our
S_{A}
has the
P
property for all
ϒ
.
Theorem 3.2.
Let
where ϒ a real number. Then

(a)SDLCP(SA,Q)has a unique solution for all Q diagonal and the solution is diagonal.

(b)SDLCP(SA,Q)has a unique solution for all Q⪯0 and the solution is
Proof
. For part(a), let
Q
=
D
, where
D
is a diagonal matrix. By Proposition 3(ii) of
[10]
, the solution set of SDLCP(
S_{A},D
) is equal to the solution set of LCP(
C, d
), where
C
=
I
−
A
◦
A
, under obvious modifications (here,
d
denotes the vector
diag
(
D
)). Since
C
is a
P
matrix (that is, the matrix with all its principal minors positive) for all
ϒ
∈
R
, the assertion follows.
For part(b), if given
Q
is diagonal, then there is a unique solution by part(a). Assume
Q
is a nondiagonal symmetric negative semidefinite matrix. Since −
Q
⪰ 0, we claim that there is a unique
X
≻ 0 such that
S_{A}
(
X
) = −
Q
. This is because, for
and
is a linear transformation from
S
^{n}
to
S
^{n}
that maps a nondiagonal positive semidefinite matrix into a positive definite matrix; and
Therefore,
Note
X
solves SDLCP(
S_{A},Q
). Now suppose there is
X
_{1}
⪰ 0 such that
Y
_{1}
:=
S_{A}
(
X
_{1}
) +
Q
⪰ 0 and
X
_{1}
Y
_{1}
= 0. Then
S_{A}
(
X
_{1}
) =
Y
_{1}
−
Q
and
because
Y
_{1}
⪰ 0. Then both
X
_{1}
and
X
are positive definite solutions and by Lemma 7,
X
_{1}
=
X
. This completes the proof.
In addition to this, in what follows, we show that not only this
S_{A}
, but also its variants
P
∈
R
^{n×n}
invertible (
X
∈
S
^{n}
), all have the
Cone

Gus
properties when
ϒ
^{2}
< 4. We achieve this by showing
S_{A}
has the socalled
P
_{2}
′property and then applying results of J. Tao
[26]
. The
P
_{2}
′property was introduced by Chandrashekaran et. al
[2]
in 2010, and J. Tao showed that (Thm 3.3
[26]
interpreted for
V
=
S
^{n}
)
L
has the
P
_{2}
′property if and only if
and all its principal subtransformations have the
Cone

Gus
properties where
P
∈
R
^{n×n}
invertible (
X
∈
S
^{n}
).
Theorem 3.3.
The transformation S_{A} given in (9) has the
P
_{2}
′
property for ϒ
^{2}
< 4.
Proof
. Let
Suppose
XS_{A}
(
X
)
X
⪯ 0. If
X
is invertible (i.e.,
X
≻ 0), then
since
X
^{−1}
is symmetric. Let
Q
:=
S_{A}
(
X
) ⪯ 0. Then by Theorem 3(b), SDLCP(
S_{A},Q
) has a unique solution and the solution is
which equals to
by linearity of
S_{A}
. But this equals −
X
which is a solution to SDLCP(
S_{A},Q
), and so 0 ≻ −
X
⪰ 0 which is a contradiction. Therefore,
X
can’t be invertible.
Now suppose det
X
= 0, i.e.,
Carrying out the algebra,
Then
XS_{A}
(
X
)
X
⪯ 0 leads to [
XS_{A}
(
X
)
X
]
_{11}
≤ 0 and [
XS_{A}
(
X
)
X
]
_{22}
≤ 0. Upon putting
in [
XS_{A}
(
X
)
X
]
_{22}
, we get
because
X
⪰ 0 and
ϒ
^{2}
< 4. Hence the last item in the above equations vanishes, therefore, we have cases
(i)
x
_{3}
= 0, or (ii) (
x
_{1}
−
x
_{3}
)
^{2}
+(4−
ϒ
^{2}
)
x
_{1}
x
_{3}
= 0. For the case (i), if
x
_{3}
= 0, then
x
_{2}
= 0. Then [
XS_{A}
(
X
)
X
]
_{11}
leads to
Hence
x
_{1}
= 0, and therefore
X
= 0. In case (ii), we get 0 ≤ (
x
_{1}
−
x
_{3}
)
^{2}
= −(4 −
ϒ
^{2}
)
x
_{1}
x
_{3}
≤ 0, which leads to [
x
_{1}
=
x
_{3}
] and [
x
_{1}
= 0 or
x
_{3}
= 0] since
ϒ
^{2}
≠ 4. We get
X
= 0 in each case. Therefore,
S_{A}
has the
P
′
_{2}
property.
Hence, for any real invertible matrix
P, L
(
X
) :=
P^{T} S_{A}
(
PXP^{T}
)
P
has only the trivial solution for any given
Q
⪰ 0 which would make
X
⪰ 0,
L
(
X
)+
Q
⪰ 0, and
tr
(
X
(
L
(
X
)+
Q
)) = 0, when
ϒ
^{2}
< 4: From the above proof, it is more or less clear that
S_{A}
is not
P
′
_{2}
if
ϒ
^{2}
> 4. When
ϒ
^{2}
= 4, the following example shows that
S_{A}
is not
P
′
_{2}
as well.
Example
Let
Then
XS_{A}
(
X
)
X
= 0 ⪯ 0, yet
X
≠ 0. Hence
S_{A}
is not
P
′
_{2}
.
4. Conclusion and Acknowledgements
In this paper, we took a baby step to characterize the
GUS
property of the Stein transformation which hasn’t been succeeded for the last decade. We hope the presentation of this paper would bring interests of many other talented mathematicians to work on this newly generated problem. We are indebted to Professor
Muddappa S. Gowda
of University of Maryland Baltimore County for the idea of the real numerical radius and helpful suggestions. We are grateful to Professor
Jiyuan Tao
of Loyola University of Maryland for pointing out right resources for the paper.
BIO
Yoon J. Song received her M.Sc. and Ph.D. at University of Maryland Baltimore County, USA. She has been at Soongsil University since 2010. Her research interests are semidefinite complementarity problems in optimization.
Department of Mathematics, Soongsil University, Seoul 156743, Korea.
email: yoonjsong@ssu.ac.kr
Seon Ho Shin is currently a visiting professor at Hongik University. She received her Ph.D at Sookmyung Women’s University. Her research interests include order structures, category theory, and mathematical aspects of cryptography.
Department of Mathematics Education, Hongik University, Seoul 121791, Korea.
email: shinsh@hongik.ac.kr
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