Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2 ()
1. Introduction
A
sign pattern is a matrix
with entries from the set
where
(respectively,
) denotes a positive (respectively, negative) real number; see, e.g., [1]. The set of all real matrices with the same sign pattern as the
sign pattern
is the qualitative class
![](https://www.scirp.org/html/1-2230020\9876a4f2-e74b-48f0-bec0-c445c30c29fa.jpg)
A subpattern of an
sign pattern
is a sign pattern
obtained by replacing some (possibly empty) subset of the nonzero entries of
with zeros. If
is a subpattern of
, then
is a superpattern of
. Two square zero-nonzero patterns
and
are equivalent if one can be obtained from the other by any combination of transposition and permutation similarity. A sign pattern
is reducible if it is permutation similar to a pattern of the form
![](https://www.scirp.org/html/1-2230020\b09cb5eb-dde2-4af6-9b5b-8577c79aea93.jpg)
where
are square and non-vacuous. A pattern is irreducible if it is not reducible.
The inertia of a matrix
is an ordered triple
where
and
are the number of eigenvalues of
with positive, negative and zero real part, respectively; see, e.g., [2]. The refined inertia of
is the ordered quadruple
of nonnegative integers that sum to
, where
is the inertia of
while
is the number of 0 as an eigenvalue of
and
is the number of eigenvalues of
with nonzero pure imaginary eigenvalues; see, e.g., [3]. The inertia (respectively, refined inertia) of a sign pattern
is
(respectively,
). A
sign pattern ![](https://www.scirp.org/html/1-2230020\9e684d2c-3ee8-490d-8bdc-0df70176b926.jpg)
is an inertially arbitrary pattern (IAP) if given any ordered triple
of nonnegative integers with
, there exists a real matrix ![](https://www.scirp.org/html/1-2230020\b6035073-5659-46cf-8566-348afc9d23cb.jpg)
such that
; see, e.g., [4,5] and the reference therein. Similarly,
is a refined inertially arbitrary pattern (rIAP) if given any ordered quadruple
of nonnegative integers that sum to
, there exists a real matrix
such that
; see, e.g., [3].
Let
be a nonempty, proper subset of the set of all inertias for any
zero-nonzero (or sign) pattern
. If
is sufficient for
to be inertially arbitrary, then
is said to be a critical set of inertias for zero-nonzero (or sign) patterns of order
and if no proper subset of
is a critical set of inertias, then
is said to be a minimal critical set of inertias for zerononzero (or sign) patterns of order
; see, e.g., [6]. Critical sets of refined inertias for irreducible zero-nonzero patterns are defined in [7]. Similarly, we introduce the concept of a critical set of refined inertias for sign patterns. Let
be a nonempty, proper subset of all refined inertias. Then,
is called a critical set of refined inertias for sign patterns of order
, if the set
is sufficient for any sign pattern
to be refined inertially arbitrary. If no proper subset of
is a critical set of refined inertias, then
is a minimal critical set of refined inertias for irreducible sign patterns of order
. We note that all minimal critical sets of inertias for irreducible sign patterns of order 2 have been identified in [6]. But, identifying all minimal critical sets of inertias for irreducible zero-nonzero (or sign) patterns of order
has been posed as an open question in [6]. Also open is the minimum cardinality of such a set. The concept of critical sets of refined inertias for sign pattern is introduced for the first time. In this work, we concentrate on the minimal critical sets of refined inertias for irreducible sign patterns of order 2.
Our work is organized as follows. Section 2 describes some preliminary results on the refined inertias of sign patterns. The minimal critical sets of refined inertias for irreducible sign patterns of order 2 are identified in Section 3. In Section 4, as a by-product, an alternative proof is given to identify all minimal critical sets of inertias for
irreducible sign patterns. Some concluding remarks are given in Section 5.
2. Preliminaries
Recall that a sign pattern
has its associated digraph
with vertex set
and for all
and
, a positive (resp., negative ) arc from
to
if and only if
(resp.,
). A (directed) simple cycle (or a k-cycle)
of length k is a sequence of k arcs
such that these vertices are distinct; see, e.g., [1]. The sign positive (or negative) of a simple cycle in a sign pattern
is the actual product of the entries in the cycle, following the obvious rules that multiplication is commutative and associative, and (+)(+) = +,(+)(−) = −.
Lemma 2.1. let
be an irreducible sign pattern of order 2. Then the following are equivalent:
(1)
is spectrally arbitrary;
(2)
is inertially arbitrary;
(3) Up to equivalence,
;
(4) Associated digraph of
, has two loops of opposite sign and a negative 2-cycle.
Proof. The equivalences of
follow from Proposition 3 in [6]. The equivalence of
can be verified directly.
It is known that there are seven refined inertias for
sign patterns. To identify all minimal critical sets of refined inertias for irreducible sign patterns of order 2, the following three sign patterns are necessary to be investigated.
Lemma 2.2. Let
Then,
allows the only refined inertias (0, 1, 1, 0), (0, 2, 0, 0) and (1, 1, 0, 0).
Proof. Since
requires every realization with a negative trace, the refined inertias (2, 0, 0, 0), (0, 0, 2, 0), (0, 0, 0, 2) and (1, 0, 1, 0) cannot be allowed by
. For the remaining refined inertias, consider realizations of
with refined inertias (0, 1, 1, 0), (0, 2, 0, 0) and (1, 1, 0, 0), respectively. It follows that
allows the only refined inertias (0, 1, 1, 0), (0, 2, 0, 0), (1, 1, 0, 0).
Lemma 2.3. Let
Then,
allows all refined inertias except (0, 1, 1, 0), (0, 2, 0, 0), (0, 0, 2, 0) and (0, 0, 0, 2).
Proof. Since
requires every realization with a positive trace, the refined inertias (0, 1, 1, 0), (0, 2, 0, 0), (0, 0, 2, 0) and (0, 0, 0, 2) cannot be allowed by
. Consider the following realizations of
,
and
with refined inertias (20, 0, 0), (1, 0, 1, 0) and (1, 1, 0, 0), respectively. It follows that
allows all refined inertias except (0, 1, 1, 0), (0, 2, 0, 0), (0, 0, 2, 0) and (0, 0, 0, 2).
Lemma 2.4. Let
Then,
allows the refined inertia (0, 0, 0, 2).
Proof. Lemma 2.4 follows from the fact that a realization of
, has (0, 0, 0, 2) as its refined inertia.
3. Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2
We are now to identify all minimal critical sets of refined inertias for irreducible sign patterns of order 2.
Theorem 3.1. The set {(0, 0, 2, 0)} is the only minimal critical set with a single refined inertia for
ireducible sign patterns.
Proof. Lemma 2.2 indicates that {(0, 2, 0, 0), {(1, 1, 0, 0)} and {(0, 1, 1, 0)} cannot be a minimal critical set of refined inertias. Lemma 2.3 indicates that {(2, 0, 0, 0)} and {(1, 0, 1, 0)} cannot be a minimal critical set of refined inertias. Lemma 2.4 indicates that {(0, 0, 0, 2)} cannot be a minimal critical set of refined inertias. So, it suffices to show that the set {(0, 0, 2, 0)} is a minimal critical sets of refined inertias.
If {(0, 0, 2, 0)} is allowed by an arbitrary irreducible sign pattern
of order 2, then all the main diagonal entries of
must be nonzero. Since
allows a realization with a zero trace, the two diagonal entries of
are of opposite sign. That is to say, the associated digraph
of
, has a positive loop and a negative loop. For
allows a realization with zero determinant,
has a negative 2-cycle. It follows from Lemma 2.1 that
is refined inertially arbitrary.
Theorem 3.2. The refined inertia sets {(0, 0, 0, 2), (1, 0, 1, 0)}, {(0, 0, 0, 2), (0, 1, 1, 0)}, {(0, 0, 0, 2), (2, 0, 0, 0)}, {(0, 0, 0, 2), (0, 2, 0, 0)}, {(0, 0, 0, 2), (1, 1, 0, 0)}, {(1, 0, 1, 0), (0, 1, 1, 0)}, {(1, 0, 1, 0), (0, 2, 0, 0)}, {(1, 0, 1, 0), (2, 0, 0, 0)} and {(2, 0, 0, 0), (0, 2, 0, 0)} are minimal critical sets of refined inertias for irreducible sign patterns of order 2.
Proof. Let
be an arbitrary irreducible sign pattern of order 2. If {(0, 0, 0, 2), (1, 0, 1, 0)}
, then
allows a realization with a positive trace and a realization with zero trace. It follows that
has a positive loop and a negative loop. Since
allows a realization with zero determinant,
has a negative 2-cycle. By Lemma 2.1,
is refined inertially arbitrary and {(0, 0, 0, 2), (1, 0, 1, 0)} is a minimal critical set of refined inertias for
irreducible sign patterns. Similarly, we can show the refined inertias {(0, 0, 0, 2), (0, 1, 0)}, {(0, 0, 0, 2), (2, 0, 0, 0)}, {(0, 0, 0, 2), (0, 2, 0, 0)}, {(1, 0, 1, 0), (0, 1, 1, 0)}, {(1, 0, 1, 0), (0, 2, 0, 0)}, {(1, 0, 1, 0), (2, 0, 0, 0)} and {(2, 0, 0, 0), (0, 2, 0, 0)} are minimal critical sets of refined inertias for
irreducible sign patterns.
For the refined inertia set {(0, 0, 0, 2), (1, 1, 0, 0)}
, we claim that the diagonal entries of
must be nonzero. In fact, assume that there exists at least a zero diagonal entry of
. Then sign pattern
requires nonsingularity. It is contradicted that
allows two realizations with a positive and negative determinant, respectively. So, the diagonal entries of
must be nonzero. And the fact that the diagonal entries of
are of opposite sign follows from that
allows a realization with a zero trace.
has a negative simple cycle of length 2 follows from that the inertia (0, 0, 0, 2)
.
Next we identify all minimal critical sets of refined inertias for
irreducible sign patterns.
Theorem 3.3. The sets {(0, 0, 2, 0)}, {(0, 0, 0, 2), (1, 0, 1, 0)}, {(0, 0, 0, 2), (0, 1, 1, 0)}, {(0, 0, 0, 2), (2, 0, 0, 0)}, {(0, 0, 0, 2), (0, 2, 0, 0)}, {(0, 0, 0, 2), (1, 1, 0, 0)}, {(1, 0, 1, 0), (0, 1, 1, 0)}, {(1, 0, 1, 0), (0, 2, 0, 0)}, {(1, 0, 1, 0), (2, 0, 0, 0)} and {(2, 0, 0, 0), (0, 2, 0, 0)} are the only minimal critical sets of refined inertias for
irreducible sign patterns.
Proof. By Theorems 3.1 and 3.2, the refined inertia sets stated in Theorem 3.3 are minimal critical sets of refined inertias for
irreducible sign patterns. To show there exists no other minimal critical sets of refined inertias, it suffices to show that the remaining nine refined inertia sets with cardinality 2, {(1, 0, 1, 0), (2, 0, 0, 0)}, {(1, 0, 1, 0), (1, 1, 0, 0)}, {(2, 0, 0, 0), (1, 1, 0, 0)}, {(0, 1, 1, 0), (0, 2, 0, 0)}, {(0, 1, 1, 0), (1, 1, 0, 0)}, {(1, 1, 0, 0), (0, 2, 0, 0)} and the two refined inertia sets with cardinality 3, {(1, 0, 1, 0), (2, 0, 0, 0), (1, 1, 0, 0)} and {(0, 1, 1, 0), (0, 2, 0, 0), (1, 1, 0, 0)} are not critical sets of refined inertias. By Lemma 2.3, {(1, 0, 1, 0), (2, 0, 0, 0)}, {(1, 0, 1, 0), (1, 1, 0, 0)}, {(2, 0, 0, 0), (1, 1, 0, 0)} and {(1, 0, 1, 0), (2, 0, 0, 0)}, (1, 1, 0, 0)} are not critical sets of refined inertias. By Lemma 2.2, {(0, 1, 1, 0), (0, 2, 0, 0)}, {(0, 1, 1, 0), (1, 1, 0, 0)}, {(1, 1, 0, 0), (0, 2, 0, 0)} and {(0, 1, 1, 0), (0, 2, 0, 0)}, (1, 1, 0, 0)} are not critical sets of refined inertias.
The following theorem follows directly for Theorem 3.3.
Theorem 3.4. Let
be a irreducible sign pattern of order 2. Then the following are equivalent:
1)
is refined inertially arbitrary;
2)
allows (0, 0, 2, 0);
3)
allows (0, 0, 0, 2) and (1, 0, 1, 0);
4)
allows (0, 0, 0, 2) and (0, 1, 1, 0);
5)
allows (0, 0, 0, 2) and (2, 0, 0, 0);
6)
allows (0, 0, 0, 2) and (0, 2, 0, 0);
7)
allows (0, 0, 0, 2) and (1, 1, 0, 0);
8)
allows (1, 0, 1, 0) and (0, 1, 1, 0);
9)
allows (1, 0, 1, 0) and (0, 2, 0, 0);
10)
allows (1, 0, 1, 0) and (2, 0, 0, 0);
11)
allows (2, 0, 0, 0) and (0, 2, 0, 0).
4. Minimal Critical Sets of Inertias for Irreducible Sign Patterns of Order 2
In [6], all minimal critical sets of inertias for
irreducible sign patterns, which are restated here as Theorem 4.1, have been identified. In this section, we present an alternative proof in terms of critial sets of refined inertias, as a by-product.
Theorem 4.1. The sets {(2, 0, 0), (0, 0, 2)}, {(0, 2, 0), (0, 0, 2)}, {(2, 0, 0), (0, 1, 1)}, {(0, 2, 0), (1, 0, 1)}, {(1, 1, 0), (0, 0, 2)}, {(1, 0, 1), (0, 0, 2)}, {(0, 1, 1), (0, 0, 2)}, {(2, 0, 0), (0, 2, 0)} and {(1, 0, 1), (0, 1, 1)} are the only minimal critical sets of inertias for
irreducible sign patterns.
Proof. Note that an irreducible
sign pattern:
allows {(0, 0, 2), (2, 0, 0)} if and only if it allows
{(2, 0, 0, 0), (0, 0, 2, 0)} or {(2, 0, 0, 0), (0, 0, 0, 2)}allows {(0, 0, 2), (0, 2, 0)} if and only if it allows
{(0, 2, 0, 0), (0, 0, 2, 0)} or {(0, 2, 0, 0), (0, 0, 0, 2)}allows {(2, 0, 0), (0, 1, 1)} if and only if it allows
{(2, 0, 0, 0), (0, 1, 1, 0)}allows {(0, 2, 0), (1, 0, 1)} if and only if it allows
{(0, 2, 0, 0), (1, 0, 1, 0)}allows {(1, 1, 0), (0, 0, 2)} if and only if it allows
{(0, 0, 0, 2), (1, 1, 0, 0)} or {(0, 0, 2, 0), (1, 1, 0, 0)}allows {(1, 0, 1), (0, 0, 2)} if and only if it allows
{(0, 0, 0, 2), (1, 0, 1, 0)} or {(0, 0, 2, 0), (1, 0, 1, 0)}allows {(0, 1, 1), (0, 0, 2)} if and only if it allows
{(0, 0, 0, 2), (0, 1, 1, 0)} or {(0, 0, 2, 0), (0, 1, 1, 0)}allows {(2, 0, 0), (0, 2, 0)} if and only if it allows
{(2, 0, 0, 0), (0, 2, 0, 0)}allows {(1, 0, 1), (0, 1, 1)} if and only if it allows
{(0, 1, 1, 0), (1, 0, 1, 0)}.
Since all of the refined inertia sets above are critical sets of refined inertias (not necessarily minimal), the above sets of inertias are critical sets, and indeed are also easily seen to be minimal (since at least one of the corresponding sets of refined inertias is minimal in each case). It follows that the above sets of inertias are the only minimal critical sets of inertias because the corresponding sets of refined inertias are the only minimal critical sets of refined inertias.
5. Concluding Remarks
We have identified all minimal critical sets of refined inertias for irreducible sign patterns of order 2. As a by-product, all minimal critical sets of inertias for irreducible sign patterns of order 2 have also been identified in a new proof. In a follow-up paper, we will consider other cases, e.g.,
, though identification of all critical sets of inertias for irreducible sign patterns of order
has been posed as an open question in [6].
6. Acknowledgements
The authors would like to express their great gratitude to the referees and editor for their constructive comments and suggestions that led to the enhancement of this paper. This research was supported in part by the Sci. & Tech. Research Fund of Huaiyin Institute of Technology (HGB1111), National Natural Science Foundation of China (11201168) and Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No. 12KJB110001).