Two n × n G-classes of matrices having finite intersection

Setareh Golshan; Ali Armandnejad; Frank J. Hall

doi:10.1515/spma-2022-0178

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Two n × n G-classes of matrices having finite intersection

Setareh Golshan , Ali Armandnejad and Frank J. Hall

Published/Copyright: November 15, 2022

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From the journal Special Matrices Volume 11 Issue 1

Abstract

Let M n be the set of all n × n real matrices. A nonsingular matrix A ∈ M n is called a G-matrix if there exist nonsingular diagonal matrices D 1 and D 2 such that A − T = D 1 A D 2 . For fixed nonsingular diagonal matrices D 1 and D 2 , let G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } , which is called a G-class. The purpose of this short article is to answer the following open question in the affirmative: do there exist two n × n G-classes having finite intersection when n ≥ 3 ?

Keywords: G-matrix; G-class; inertia of matrices

MSC 2010: Primary: 15B10; Secondary: 15A30

1 Introduction

All matrices in this note have real number entries. Let M n be the set of all n × n real matrices. A nonsingular matrix A ∈ M n is called a G-matrix if there exist nonsingular diagonal matrices D 1 and D 2 such that A − T = D 1 A D 2 , where A − T denotes the transpose of the inverse of A . These matrices form a rich class and were originally studied in [1] by Fiedler and Hall; they include the orthogonal and J-orthogonal matrices. For a survey of the basic properties of G-matrices and connections to other classes of matrices, the reader can refer to [1] and [2] and references therein. Here, we just mention two other connections.

Cauchy matrices have the form C = [ c i j ] , where c i j = 1 x i + y j for some numbers x i and y j . We shall restrict to square, say n × n , Cauchy matrices – such matrices are defined only if x i + y j ≠ 0 for all pairs of indices i and j , and it is well known that C is nonsingular if and only if all the numbers x i are mutually distinct and all the numbers y j are mutually distinct. It turns out that by observation of Fiedler [3], every nonsingular Cauchy matrix is a G-matrix. So, in particular, G-matrices arise naturally as very well-defined structured nonsingular Cauchy matrices. Furthermore, G-matrices arise also in the context of “combined matrices” C ( A ) = A ∘ A − T , where ∘ denotes the Hadamard product, see [3]. For example, if A is a G-matrix, then C ( A ) = A ∘ ( D 1 A D 2 ) = D 1 ( A ∘ A ) D 2 ; so if say D 1 and D 2 are nonnegative, then C ( A ) is nonnegative. The combined matrices appear in the chemical literature where they represent the relative gain array [4].

For fixed nonsingular diagonal matrices D 1 and D 2 , let the class of n × n G-matrices be

G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } .

We call such a class of matrices a G-class of matrices.

In this note, it is shown that for every n , there exist two n × n G-classes having finite, nonempty intersection. This answers an open question in [2] in the affirmative.

We note that the nonsingular diagonal matrices D 1 and D 2 satisfying A − T = D 1 A D 2 are, in general, not uniquely determined as we can multiply one of them by a nonzero real number and divide the other by the same number. On the other hand, for nonsingular n × n diagonal matrices D 1 and D 2 , the following known result from [1] shows that if A − T = D 1 A D 2 , then D 1 and D 2 have the same inertia matrix. For the definitions of the inertia and the corresponding inertia matrix of a general Hermitian matrix, the reader can refer to [5, pp. 281–282]. Simply put, the inertia matrix of a Hermitian matrix A is the diagonal matrix

diag ( 1 , … , 1 , − 1 , … , − 1 , 0 , … , 0 ) ,

where the number of 1 ′ s , − 1 ′ s , and 0 ′ s is the number of positive, negative, and zero eigenvalues, respectively, of A .

Proposition 1.1

Suppose A is a G-matrix and A − T = D 1 A D 2 , where D 1 and D 2 are nonsingular diagonal matrices. Then, the inertia of D 1 is equal to the inertia of D 2 .

2 Solution of the open question

By a signature matrix, we mean a diagonal matrix where each diagonal entry is ± 1 . Let D be a nonsingular diagonal matrix with the inertia matrix J (a signature matrix having all its positive entries in the upper left corner). Then, there exists a permutation matrix P such that D = ∣ D ∣ P T J P , where ∣ D ∣ is obtained by taking the absolute value on entries of D .

For a fixed signature matrix J , Γ n ( J ) = { A ∈ M n : A ⊤ J A = J } . In fact,

Γ n ( J ) = G ( J , J ) .

We mention that the matrices in Γ n ( J ) are precisely the J-orthogonal matrices discussed in [6, 7,8,9]. Also, note that when J is I or − I , Γ n ( J ) = O n , the set of all n × n orthogonal matrices.

In [2], the authors proved the following theorem (Theorem 2.2 of [2]).

Theorem 2.1

Let D 1 and D 2 be nonsingular diagonal matrices with the same inertia matrix J. Then, there exist permutation matrices P and Q such that

G ( D 1 , D 2 ) = { ∣ D 1 ∣ − 1 ∕ 2 P T A Q ∣ D 2 ∣ − 1 ∕ 2 : A ∈ Γ n ( J ) } .

This characterization shows that G ( D 1 , D 2 ) is in fact nonempty.

Finally, we mention one other preliminary result.

Theorem 2.2

[2, Theorem 3.1] Let D 1 , D 2 , D 3 , and D 4 be nonsingular diagonal matrices, all of which have the same inertia matrix I or − I . Then,

G ( D 1 , D 2 ) = G ( D 3 , D 4 )

if and only if there exists a positive number d such that D 3 = d D 1 and D 4 = 1 d D 2 .

With that background, we can now answer the open question. We remark that an example is already given in [2] in the case when n = 2 . However, from that example, no inductive procedure is apparent. But now we are able to give patterns in D 1 , D 2 , D 3 , and D 4 that are amenable to induction.

Theorem 2.3

There exist two n × n G-classes having finite, nonempty intersection when n ≥ 3 .

Proof

Let

D 1 = diag 1 , 1 2 , 1 3 , … , 1 n , D 2 = diag ( 1 , 2 , 3 , … , n ) , D 3 = diag 1 3 , 1 4 , … , 1 n + 2 , D 4 = diag ( 3 , 4 , … , n + 2 ) .

The inertia matrix of each of D 1 , D 2 , D 3 , and D 4 is I . By using Theorem 2.2,

G ( D 1 , D 2 ) ≠ G ( D 3 , D 4 ) .

Let A ∈ G ( D 1 , D 2 ) ∩ G ( D 3 , D 4 ) . Since here the inertia matrix of each D i is J = I , Γ n ( J ) = O n , and the permutation matrices P and Q are not needed; therefore, by using Theorem 2.1, there are V , W ∈ O n such that

A = D 1 − 1 2 V D 2 − 1 2 = D 3 − 1 2 W D 4 − 1 2 ,

which implies that

W = D 3 1 2 D 1 − 1 2 V D 2 − 1 2 D 4 1 2

(Since W ∈ O n , this means that also D 3 1 ∕ 2 D 1 − 1 ∕ 2 V D 2 − 1 ∕ 2 D 4 1 ∕ 2 ∈ O n .).

From W = D 3 1 2 D 1 − 1 2 V D 2 − 1 2 D 4 1 2 , with W = [ w i j ] and V = [ v i j ] , it follows that

w i j = i i + 2 j + 2 j v i j .

For all i < j , we have 0 < i i + 2 j + 2 j < 1 , and consequently, when v i j = 0 , w i j = 0 , and v i j ≠ 0 , w i j 2 < v i j 2 .

From the diagonal entries of W W T = I , we obtain for 1 ≤ i ≤ n ,

1 = ( W W T ) i i = ∑ j = 1 n w i j 2 = ∑ j = 1 n i ( j + 2 ) ( i + 2 ) j v i j 2 . ( ∗ i )

From the entries of V V T = I , we obtain for 1 ≤ i ≤ n ,

1 = ( V V T ) i i = ∑ j = 1 n v i j 2 , ( ∗ ∗ i )

and for each i and t with 1 ≤ i ≠ t ≤ n ,

0 = ( V V T ) i , t = ∑ j = 1 n v i j v t j . ( ∗ ∗ ∗ i , t ) .

Now, we show that the off-diagonal entries of row 1 and column 1 of V are zero. In ( ∗ 1 ) , if at least one of v 1 j ≠ 0 ( j = 2 , … , n ) , then the right-hand sides of ( ∗ 1 ) and ( ∗ ∗ 1 ) are not equal, which is a contradiction. Therefore, v 1 j = 0 , ( j = 2 , … , n ) and v 1 1 = ± 1 . Now, relations ( ∗ ∗ ∗ 1 , t ) ( 1 < t ≤ n ) imply v 2 1 = v 3 1 = ⋯ = v n 1 = 0 .

So far, we have:

V = ± 1 0 … 0 0 ⋮ ⋆ 0 .

The case where i = 2 uses the above structure of V and proceeds similar to the case where i = 1 . We arrive at

V = ± 1 0 0 … 0 0 ± 1 0 … 0 0 0 ⋮ ⋮ ⋆ 0 0 .

The induction hypothesis is that all the off-diagonal entries in V in the first k − 1 rows and columns are zero, and each diagonal entry is ± 1 . Since v k 1 , v k 2 , … , v k , k − 1 are zero, in ( ∗ k ) , if at least one of v k j ≠ 0 , ( j = k + 1 , … , n ) , then the right-hand sides of ( ∗ k ) and ( ∗ ∗ k ) are not equal, which is a contradiction. Therefore, v k , k + 1 = v k , k + 2 = … = v k n = 0 , and so v k k = ± 1 . Now, relations ( ∗ ∗ ∗ k , t ) ( k < t ≤ n ) imply v k + 1 , k = v k + 2 , k = ⋯ = v n k = 0 . So, the off-diagonal entries of row k and column k of V are zero. Thus, by induction, V = diag ( ± 1 ) , and hence

A = D 1 − 1 2 V D 2 − 1 2 = diag ( ± 1 ) .

So A can only be of the form diag ( ± 1 ) . Therefore, the intersection of G ( D 1 , D 2 ) and G ( D 3 , D 4 ) is finite, and it has 2 n matrices.□

Acknowledgment

The authors wish to express their appreciation to the referees for a careful reading of the article and the valuable suggestions.

Funding information: The authors have no internal or external funding support.
Conflict of interest: The authors have no conflict of interest to report. The author Frank J. Hall is on the Editorial Advisory Board of the Special Matrices journal, but this did not affect the final decision for the article.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-06-14

Revised: 2022-10-09

Accepted: 2022-10-18

Published Online: 2022-11-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/spma-2022-0178

Keywords for this article

G-matrix; G-class; inertia of matrices

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