INVERSE PRINCIPALLY CENTROGONAL MATRICES

LeRoy B. Beasley1, Sang-Gu Lee2 §, Han-Guk Seol3

1Department of Mathematics and Statistics
Utah State University
Logan, UT 84322-3900, USA
e-mail: lbeasley@math.usu.edu
2,3Department of Mathematics
College of Science
Sung Kyun Kwan University
Suwon 440-740, SOUTH KOREA
2e-mail: sglee@math.skku.ac.kr
3e-mail: shk@math.skku.ac.kr
Abstract: A real nonsingular n × n matrix A = (aij) is called centrogonal if
A−1 = (an+1−i,n+1−j), it is called principally centrogonal if all leading principal
submatrices of A are centrogonal, and it is called inverse principally centrogonal
if A−1 is principally centrogonal. We give a necessary and sufficient condition
for a principally centrogonal matrix to be an inverse principally centrogonal
matrix.
AMS Subject Classification: 15A03, 15A04, 15A33
Key Words: principally centrogonal matrix, matrix rotation, centrosymmet-
ric
1. Introduction
A real nonsingular n × n matrix A = (aij) is called centrogonal if A−1 =
(an+1−i,n+1−j). A centrogonal matrix is called principally centrogonal if all
leading principal submatrices of A are centrogonal,and a principally centrog-onal matrix, A, is called inverse principally centrogonal if A−1 is principally
centrogonal. Centrogonal matrices were examined in [1], where motivation for
the study is given relating them to persymmetric (AT = AR) and centrosymmetric
(AR = A) matrices. Also, centrogonal matrices are related with bino-
mial coefficients and principal centrogonality of the matrix A, when suitably
normalized, shares another significant property with the unit matrix: It has
the same characteristic polynomial. If A is a centrogonal matrix then A−1 is
centrogonal, but if A is a principally centrogonal matrix, A−1 is not neces-
sarily principally centrogonal. In this article we investigate the properties of
inverse principally centrogonal matrices. We will give necessary and sufficient
condition for a principally centrogonal matrix to be an inverse principally cen-
trogonal matrix. In analogy to the fact that the result of transposing a matrix
is called the transpose, we call the result of rotating a matrix the rotate [1].
Thus, for a square matrix A = (aij)1 i,j n the rotate, denoted AR, is defined
by AR = (an+1−i,n+1−j)1 i,j n. Clearly, for the rotate AR of A we have that
AR = JAJ, where J is the exchange matrix (or a per-identity matrix), i.e.,
J =
0BBBBB@
0 0 · · · 0 1
0 0 · · · 1 0
. . .
. . .
. . .
. . .
. . .
0 1 · · · 0 0
1 0 · · · 0 0
1CCCCCA
.
For a vector a = (a1, a2, · · · , an)T the lower triangular n × n matrix B(a)
is defined by
B(a) =
0BBB@
a1 0 · · · 0
a2 a1 · · · 0
...
...
. . .
...
an an−1 · · · a1
1CCCA
.
Clearly, for the rotate AR of A we have that AR = JAJ. The following prop-
erties of the rotation operator R was shown in [1]:
(AT )R = (AR)T ,
(AR)R = A,
(A−1)R = (AR)−1 ,2. Inverse Principally Centrogonal Matrices
The properties of centrogonal matrices were examined in [1]. In this paper,
we give some conditions for the inverse principally centrogonal matrices. In
[1], it was shown that a nonsingular matrix A is centrogonal if and only if
there exist an 2 {−1, 1}, a nonsingular matrix B 2 Rn×n and a symmetric
n × n permutation matrix P such that A = B−1PBJ. In particular, the
matrix A−1AR is a centrogonal matrix if 2 {−1, 1} and A is nonsingular.
A further specialization yields that B−1BT is centrogonal if 2 {−1, 1} and
B is nonsingular and persymmetric, i.e. B = JBT J. Since Toeplitz matrices
are persymmetric, B−1BT is centrogonal if B is a nonsingluar Toeplitz matrix
because of this property.
Principal centrogonality is, of course, a strong condition. Nevertheless, it
might be suprising that a matrix is completely determined by its first row.
Theorem 2.1. [1] Let A = (aij) be a nonsingluar n × n matrix, a =
(a11, a12, . . . , a1n)T and B = B(a). Then A is principally centrogonal if and
only if A = a11B−1BT and a11 2 {−1, 1}.
Example 2.1. Let
A =0
@
1 1 1
−1 0 0
0 −1 0
1A
.
Then A is a principally centrogonal matrix. But
A−1 = AR =0
@
0 −1 0
0 0 −1
1 1 1
1A
is not principally centrogonal.
Example 2.2. Let
A = a11 a12
a21 a22
be a 2×2 principally centrogonal matrix. Then by Theorem 2.1, A = a11B−1BT ,
where B = a11 0
a12 a11 and a11 2 {−1, 1}. So,
A = a11B−1BT = 1 a12
−a12 1 − a2or −1 a12
−a12 a2
12 − 1 . If A is an inverse principally centrogonal matrix, then
A−1 = AR has one of the forms
1 − a2
12 −a12
a12 1 or a2
12 − 1 −a12
a12 −1 .
Since A is a centrogonal matrix, we also have
A−1 = AR = a22 a21
a12 a11 .
So we have the following:
a22 a21
a12 a11 = 1 − a2
12 −a12
a12 1 or a2
12 − 1 −a12
a12 −1
As the same manner, ˆB −1 ˆB T = 1 a21
−a21 1 − a2
21 or −1 a21
−a21 a2
21 − 1 because
ˆB
= a22 0
a21 a22 . We have the following:
1 − a2
12 −a12
a12 1 = 1 a21
−a21 1 − a2
21 or −1 a21
−a21 a2
21 − 1 ,
a2
12 − 1 −a12
a12 −1 = 1 a21
−a21 1 − a2
21 or −1 a21
−a21 a2
21 − 1 .
Hence we find that all the 2 × 2 inverse principally centrogonal matrices are:
1 0
0 1 , −1 0
0 −1 , 1 p2
−p2 −1 ,
1 −p2 p2 −1 , −1 p2
−p2 1 , −1 −p2 p2 1 .
We now consider inverse principally centrogonal matrices of order n > 2.
Let A = (aij) be a principally centrogonal matrix. Then by Theorem 1,
A = a11B−1BT and a11 2 {−1, 1}. Since A is centrogonal, AR = A−1 and
centrogonal. So, if we assume that AR is principally centrogonal, then A is in-
verse principally centrogonal. Hence, we have the condition that A−1 = AR =
(a11B−1BT )R = a11(BR)−1(BR)T .
Lemma 2.1. Let A = (aij) be an n×n principally centrogonal matrix. If
a11 = ann 2 {−1, 1} and a1k = 0 = ans for k = 2, 3, . . . , n and s = 1, 2, . . . , n−1,
then A is inverse principally centrogonal.
Proof. Since A is principally centrogonal, by Theorem 1, A =
a11B−1BT , where B = a11In. So, A = I or −I and A−1 = AR = A.
Theorem 2.2. Let A = (aij) be an n × n principally centrogonal matrix.
If a11 = a12 = · · · = a1n, then the matrix A is the companion matrix whose
characteristic polynomial is xn − xn−1 + · · · + (−1)n.
Proof. Let A = [aij ] be a principally centrogonal matrix of order n. By
Theorem 1, A = a11B−1BT , where B = B(a) and a11 2 {−1, 1}. Since the
entries of A have the property that a11 = a12 = · · · = a1n, so B = (bij), where
bij = a11, (i j) and bij = 0, otherwise. Hence A = a11B−1BT = C or − C,
where the matrix have the form;
C =
0BBBBB@
1 1 · · · 1
−1 0 · · · 0
0 −1
...
0
0 0 −1 0
1CCCCCA
.
Theorem 2.3. Let A = (aij) be an n × n principally centrogonal matrix.
If the matrix A is symmetric, then A = diag (d1, d2, . . . , dn), where di = +1 for
all i or di = −1 for all i.
Proof. Let A = [aij ] be a principally centrogonal matrix of order n. By
Theorem 1, A = a11B−1BT , where B = B(a) and a11 2 {−1, 1}. Since A is
symmetric A = AT and thus a11B−1BT = a11(BT )T (BT )−1 = a11B(BT )−1. So
B2 = (BT )2. Since B = B(a) is lower triangular and B2 is also lower triangular,
we have that B2 is a diagonal matrix. Thus, B = diag (a11, a11, . . . , a11).
Example 2.3. There are nontivial inverse pincipally centrogonal matrices,
for example, let
F =0
@
1 2 2
−2 −3 −2
2 2 1
1A
.
It can be easily checked that F is an inverse principally centrogonal matrix.
Theorem 2.4. Let A = (aij) be an n × n principally centrogonal matrix
with decomposition A = a11B−1BT and a11 = ann 2 {−1, 1}, where B = B(a).
Suppose that ˆB = B(ˆa), where ˆa = (ann, ann−1, . . . ,
an1). Then A is inverse principally centrogonal matrix if and only if the following
matrix equation holds:
B−1BT J = ±J ˆB −1 ˆB T
Proof. Let A be an n × n inverse principally centrogonal matrix with de-
composition A = a11B−1BT and a11 = ann 2 {−1, 1}, where B = B(a). Since
A is principally centrogonal, by Theorem 1 and properties of the rotate R,
A−1 = AR = a11(B−1BT )R = a11J(B−1BT )J,
where a11 2 {−1, 1}. Also, since A is inverse principally centrogonal, A−1 is
principally centrogonal matrix. Hence A−1 = ann ˆB−1 ˆB T , where ann 2 {−1, 1}.
Since a11, ann 2 {−1, 1}, a11 = ann or a11 = −ann. It follows that
(B−1BT )R = ±ˆB −1 ˆB T ,
or
B−1BT J = ±J ˆB−1 ˆBT .
Conversely, if we assume that hold the matrix equation B−1BT J = ±J ˆB−1ˆB T .
Since A is principally centrogonal matrix A−1 = AR = a11(B−1BT )R = a11J(B−1BT )J.
But since B−1BT J = ±J ˆB−1 ˆB T , A−1 = ±a11 ˆB −1 ˆB T . This means that A−1 is
a principally centrogonal matrix. The proof is complete.
Acknowledgements
This paper was supported in part by Com2MaC-KOSEF and Korea Research
Foundation Grant KRF 2000-015-DP0005.
References
[1] Olaf Krafft, Martin Schaefer, Centrogonal matrices, Linear Algebra Appl.,
306 (2000), 145-154.
[2] G.H. Gloub, C.F. van Loan, Matrix Computations: Combinatorial and
Graph-Theoretic Problem, Second Edition, Johns Hopkins University
Press, Baltimore (1989).

(AB)R = ARBR.