Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

Keywords:
leading principal submatrix;
submatrix constraint;
generalized centrosymmetric matrix;
left and right inverse eigenpairs;
optimal approximation

Throughout this article we use some notations as follows. Let *C* ^{ n×m } be the set of all *n* × *m* complex matrices, *R* ^{ n×m } be the set of all *n* × *m* real matrices, *C* ^{ n } = *C* ^{ n×1}, *R* ^{ n } = *R* ^{ n×1}, *R* denote the set of all real numbers, *OR* ^{ n×n } denote the set of all *n* × *n* orthogonal matrices, *R*(*A*), *A* ^{ T }, *r*(*A*), tr(*A*) and *A* ^{+} be the column space, the transpose, rank, trace and the Moore–Penrose generalized inverse of a matrix *A*, respectively. *I* _{ n } is the identity matrix of size *n*. Let *e* _{ i } be the *i*th column of *I* _{ n }, and set *J* _{ n } = (*e* _{ n },…,*e* _{1}). For *A*, *B* ∈ *R* ^{ n×m }, 〈*A*, *B*〉 = tr(*B* ^{ T } *A*) denotes the inner product of matrices *A* and *B*. The induced matrix norm is called the Frobenius norm, i.e.
*R* ^{ n×m } is a Hilbert inner product space.

Generally, the left and right inverse eigenpairs problem is as follows: given partial left and right eigenpairs (eigenvalue and corresponding eigenvector) (*γ* _{ j }, *y* _{ j }), *j* = 1,…,*l*; (*λ* _{ i }, *x* _{ i }), *i* = 1,…,*h*, and a special *n* × *m* matrix set *S*, to find *A* ∈ *S* such that

(1.1)
{
A
x
i
=
λ
i
x
i
,
i
=
1,
…
,
h
,
y
j
T
A
=
γ
j
y
j
T
,
j
=
1,
…
,
l
,

where
(1.2)
{
A
X
=
X
Λ
,
Y
T
A
=
Γ
Y
T
.

This problem, which mainly arises in perturbation analysis of matrix eigenvalue and in recursive matters, has some practical applications in economic and scientific computation fields [
1,
2,
3].
Many important results have been achieved on this problem associated with many kinds of matrix sets. Li et al. [4,5,6,7,8,9] have solved the left and right inverse eigenpairs problems for skew-centrosymmetric matrices, generalized centrosymmetric matrices, *κ*-persymmetric matrices, symmetrizable matrices, orthogonal matrices and *κ*-Hermitian matrices by using the special properties of eigenpairs of matrix. Zhang and Xie [10], Ouyang [11], Liang and Dia [12] and Yin and Huang [13] have, respectively, solved the left and right inverse eigenvalue problems for real matrices, semipositive subdefinite matrices, generalized reflexive and anti-reflexive matrices and (*R*,*S*) symmetric matrices with the special structure of matrix.

Arav et al. [2] and Loewy and Mehrmann [3] studied the recursive inverse eigenvalue problem which arises in the Leontief economic model. Namely, given eigenvalue *λ* _{ i } of *A* _{ i }, in which *A* _{ i } is the *i*th leading principal submatrix of *A*, corresponding left eigenvector *y* _{ i } and right eigenvector *x* _{ i } of *λ* _{ i }, construct a matrix *A* ∈ *C* ^{ n×m } such that

Let *κ* be a real fixed product of disjoint transpositions and *J* be the associated permutation matrix. *A* = (*a* _{ ij }) ∈ *R* ^{ n×m }, if *a* _{ ij } = *a* _{ κ(i)κ(j)} (or *a* _{ ij } = −*a* _{ κ(i)κ(j)}), then *A* is called the generalized centrosymmetric matrix (or generalized centro-skewsymmetric matrix), and GCSR^{ n×m } (or GCSSR^{ n×n }) denote the set of all generalized centrosymmetric matrices (or the set of all generalized centro-skewsymmetric matrices).

From Definition 1, it is easy to derive the following conclusions.

- (1)
*J*^{ T }=*J*and*J*^{2}=*I*_{ n }. Real matrices and centrosymmetric matrices are the special cases of generalized centrosymmetric matrices with*κ*(*i*) =*i*and*κ*(*i*) =*n*−*i*+ 1 or*J*=*I*_{ n }and*J*=*J*_{ n }, respectively. - (2)
*A*∈ GCSR^{ n×n }if and only if*A*=*JAJ*and*A*∈ GCSSR^{ n×n }if and only if*A*= −*JAJ*. - (3)
*R*^{ n×n }= GCSR^{ n×n }⊕ GCSSR^{ n×n }, where the notation*V*_{1}⊕*V*_{2}stands for the orthogonal direct sum of linear subspaces*V*_{1}and*V*_{2}.

Centrosymmetry, persymmetry and symmetry are three important symmetric structures of a square *n* × *n* matrix and have profound applications, such as engineering, statistics and so on [14,15,16]. There are many meaningful results about the inverse problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint. Peng et al. [17] and Bai [18] discussed the inverse problem and the inverse eigenvalue problem of centrosymmetric matrices with a principal submatrix constraint, respectively. Zhao et al. [19] studied least squares solutions to *AX* = *B* for symmetric centrosymmetric matrices under a central principal submatrix constraint and the optimal approximation. The matrix inverse problem (or inverse eigenvalue problem) with a submatrix constraint is also called the matrix extension problem. Since de Boor and Golub [20] first put forward and considered the Jacobi matrix extension problem in 1978, many authors have studied the matrix extension problem and a series of meaningful results have been achieved [17,18,19,21,22,23,24,25,26].

Assume (*λ* _{ i },*x* _{ i }), *i* = 1,…,*m*, be right eigenpairs of *A*; (*μ* _{ i },*y* _{ i }), *j* = 1,…,*h*, be left eigenpairs of *A*. Let *X* = (*x* _{1},…,*x* _{ m }) ∈ *C* ^{ n×m }, *Λ* = diag(*λ* _{1},…,*λ* _{ m }) ∈ *C* ^{ m×m }; *Y* = (*y* _{1},…,*y* _{ h }) ∈ *C* ^{ n×h }, *Γ* = diag(*μ* _{1},…,*μ* _{ h }) ∈ *C* ^{ h×h }. The problems studied in this article can be described as follows.

Given *X* = (*x* _{1},…,*x* _{ m }) ∈ *C* ^{ n×m }, *Y* = (*y* _{1},…,*y* _{ h }) ∈ *C* ^{ n×h }, *Λ* = diag(*λ* _{1},…,*λ* _{ m }) ∈ *C* ^{ m×m }, *Γ* = diag(*μ* _{1},…,*μ* _{ h }) ∈ *C* ^{ h×h }, *A* _{0} ∈ *R* ^{ p×p }, *m* ≤ *n*, *h* ≤ *n*, *p* ≤ *n*, find *A* ∈ GCSR^{ n×n } such that

Given *A** ∈ *R* ^{ n×n }, find

This article is organized as follows. In Section 2, we first study the special properties of eigenpairs and the structure of generalized centrosymmetric matrices. Then, we provide the solvability conditions for and the general solutions of Problem I. In Section 3, we first attest the existence and uniqueness theorem of Problem II and then present the unique approximation solution with the orthogonal invariance of the Frobenius norm. Finally, we provide an algorithm to compute the unique approximation solution. Some conclusions are provided in Section 4.

Let *x* ∈ *C* ^{ n }. If *Jx* = *x* (or *Jx* = −*x*), then *x* is called the generalized symmetric (or generalized skew-symmetric) vector. Denote the set of all generalized symmetric (or generalized skew-symmetric) vectors by GC^{ n } (or GSC^{ n }).

Denote

(2.1)
P
1
=
K
1
K
1
T
,
P
2
=
K
2
K
2
T
.

(2.2)
J
=
K
1
K
1
T
−
K
2
K
2
T
=
K
(
I
n
−
r
0
0
I
r
)
K
T
.

Combining conclusion (2) of Definition 1, (
2.1) and (
2.2), it is easy to derive the following lemma.
*A* ∈ *GCSR* ^{ n×n } *if and only if*

If *J* = *J* _{ n } and *n* = 2*k*, then

If *J* = *J* _{ n }, and *n* = 2*k* + 1, then

[27] (1) *If A* ∈ *CSR* ^{2k×2k } *, then A can be written as*

(2) *If A* ∈ *CSR* ^{(2k+1)×(2k+1)} *, then A can be written as*

For a real matrix *A* ∈ *R* ^{ n×m }, its complex right eigenpairs are conjugate pairs. That is, if
*Ax* = *ax* − *by* and *Ay* = *bx* + *ay*, i.e.,

Let *A* ∈ GCSR^{ n×n }, if *Ax* = *λx*, where *λ* is a number, *x* ∈ *C* ^{ n }, and *x* ≠ 0, then we have

(2.3)
A
(
x
,
y
)
=
(
x
,
y
)
(
a
b
−
b
a
)
.

According to conclusion (2) of Definition 1, we have
(2.4)
A
J
(
x
,
y
)
=
J
(
x
,
y
)
(
a
b
−
b
a
)
.

Combining (
2.3) and (
2.4) implies
(2.5)
X
=
(
X
1
,
X
2
)
∈
R
n
×
m
,
X
1
=
J
X
1
∈
R
n
×
m
1
,
X
2
=
−
J
X
2
∈
R
n
×
(
m
−
m
1
)
,
Y
=
(
Y
1
,
Y
2
)
∈
R
n
×
h
,
Y
1
=
J
Y
1
∈
R
n
×
h
1
,
Y
2
=
−
J
Y
2
∈
R
n
×
(
h
−
h
1
)
,

[6] *If X, Λ, Y, Γ are given by* (2.5)*, then (AX* = *XΛ,Y* ^{ T } *A* = *ΓY* ^{ T } *) has a solution in GCSR* ^{ n×n } *if and only if*

(2.6)
Y
1
T
X
1
Λ
1
=
Γ
1
Y
1
T
X
1
,
X
1
Λ
1
=
X
1
Λ
1
X
1
+
X
1
,
Y
1
Γ
1
=
Y
1
Γ
1
Y
1
+
Y
1
,

(2.7)
Y
2
T
X
2
Λ
2
=
Γ
2
Y
2
T
X
2
,
X
2
Λ
2
=
X
2
Λ
2
X
2
+
X
2
,
Y
2
Γ
2
=
Y
2
Γ
2
Y
2
+
Y
2
.

*Moreover, its general solution can be expressed as*

(2.8)
A
=
A
10
+
E
F
G
,
∀
F
∈
G
C
S
R
n
×
n
,

(2.9)
A
10
=
X
1
Λ
1
X
1
+
+
(
Y
1
T
)
+
Γ
1
Y
1
T
(
I
n
−
X
1
X
1
+
)
+
X
2
Λ
2
X
2
+
+
(
Y
2
T
)
+
Γ
2
Y
2
T
(
I
n
−
X
2
X
2
+
)
∈
G
C
S
R
n
×
n
,

(2.10)
E
=
I
n
−
Y
1
Y
1
+
−
Y
2
Y
2
+
∈
G
C
S
R
n
×
n
,
G
=
I
n
−
X
1
X
1
+
−
X
2
X
2
+
∈
G
C
S
R
n
×
n
.

Combining Lemmas 1 and 3, it is easy to prove that *A* in (2.8) can be expressed as

(2.11)
A
=
K
(
A
110
+
E
1
F
1
G
1
0
0
A
210
+
E
2
F
2
G
2
)
K
T
,

where
Denote

(2.12)
(
I
p
,
0
)
K
1
E
1
=
E
¯
1
,
(
I
p
,
0
)
K
2
E
2
=
E
¯
2
,
G
1
K
1
T
(
I
p
,
0
)
T
=
G
¯
1
,
G
2
K
2
T
(
I
p
,
0
)
T
=
G
¯
2
,
A
0
−
(
I
p
,
0
)
K
1
A
110
K
1
T
(
I
p
,
0
)
T
−
(
I
p
,
0
)
K
2
A
210
K
2
T
(
I
p
,
0
)
T
=
A
¯
0
.

Combining (
2.11) and (
2.12),
(2.13)
E
¯
1
F
1
G
¯
1
+
E
¯
2
F
2
G
¯
2
=
A
¯
0
.

Suppose that the generalized singular value decomposition (GSVD) of matrix pairs

(2.14)
E
¯
1
T
=
Q
1
Σ
1
S
T
,
E
¯
2
T
=
Q
2
Σ
2
S
T
,

where
(2.15)
Σ
1
=
(
I
r
1
0
0
0
0
Θ
1
0
0
0
0
0
0
)
,
Σ
2
=
(
0
0
0
0
0
Θ
2
0
0
0
0
I
s
1
−
r
1
−
t
1
0
)
r
1
t
1
s
1
−
r
1
−
t
1
p
−
s
1
r
1
t
1
s
1
−
r
1
−
t
1
p
−
s
1

with
Suppose that the GSVD of matrix pairs

(2.16)
G
¯
1
=
P
1
Σ
3
W
T
,
G
¯
2
=
P
2
Σ
4
W
T
,

where
(2.17)
Σ
3
=
(
I
r
1
0
0
0
0
Θ
3
0
0
0
0
0
0
)
,
Σ
4
=
(
0
0
0
0
0
Θ
4
0
0
0
0
I
s
2
−
r
2
−
t
2
0
)
r
2
t
2
s
2
−
r
2
−
t
2
p
−
s
2
r
2
t
2
s
2
−
r
2
−
t
2
p
−
s
2

with
Combining (2.14) and (2.16) implies that (2.13) can be written as

(2.18)
Σ
1
T
Q
1
T
F
1
P
1
Σ
3
+
Σ
2
T
Q
2
T
F
2
P
2
Σ
4
=
S
−
1
A
¯
0
W
−
T
.

Partition
(2.19)
Q
1
T
F
1
P
1
=
(
F
111
F
112
F
113
F
121
F
122
F
123
F
131
F
132
F
133
)
,
Q
2
T
F
2
P
2
=
(
F
211
F
212
F
213
F
221
F
222
F
223
F
231
F
232
F
233
)
,

(2.20)
S
−
1
A
¯
0
W
−
T
=
(
A
011
A
012
A
013
A
014
A
021
A
022
A
023
A
024
A
031
A
032
A
033
A
034
A
041
A
042
A
043
A
044
)
.

Combining (
2.19) and (
2.20) implies that (
2.18) can be written as
(2.21)
(
F
111
F
112
Θ
3
0
0
Θ
1
F
121
Θ
1
F
122
Θ
3
+
Θ
2
F
222
Θ
4
Θ
2
F
223
0
0
F
232
Θ
4
F
233
0
0
0
0
0
)
=
(
A
011
A
012
A
013
A
014
A
021
A
022
A
023
A
024
A
031
A
032
A
033
A
034
A
041
A
042
A
043
A
044
)
.

Combining Lemma 3 and (
2.11)–(
2.21) derives the following theorem.
*If X*, *Λ*, *Y*, *Γ are given by* (2.5) *and given A* _{0} ∈ *R* ^{ p×p } *, then Problem I has a solution in GCSR* ^{ n×n } *if and only if* (2.6), (2.7) *and the following equations hold:*

(2.22)
A
013
=
0
,
A
014
=
0
,
A
024
=
0
,
A
031
=
0
,
A
034
=
0
,
A
041
=
0
,
A
042
=
0
,
A
043
=
0
,
A
044
=
0
.

(2.23)
A
=
K
(
A
110
+
E
1
F
1
G
1
0
0
A
210
+
E
2
F
2
G
2
)
K
T
,

(2.24)
F
1
=
Q
1
(
A
011
A
012
Θ
3
−
1
F
113
Θ
1
−
1
A
021
Θ
1
−
1
(
A
022
−
Θ
2
F
222
Θ
4
)
Θ
3
−
1
F
123
F
131
F
132
F
133
)
P
1
T
,

(2.25)
F
2
=
Q
2
(
F
211
F
212
F
213
F
221
F
222
Θ
2
−
1
A
023
F
231
A
032
Θ
4
−
1
A
033
)
P
2
T

From (2.23), it is easy to prove that the solution set *S* _{ E } of Problem I is a nonempty closed convex set if Problem I has a solution in GCSR^{ n×n }. We claim that for any given *A** ∈ *R* ^{ n×n }, there exists a unique optimal approximation for Problem II.

Combining (2.8)–(2.11) and Lemma 1, (2.23) can be written as

(3.1)
A
=
A
10
+
E
F
G
,
F
=
K
(
F
1
0
0
F
2
)
K
T
,

where
According to conclusion (3) of Definition 1, for any *A** ∈ *R* ^{ n×n }, there exist only one

(3.2)
A
⁎
=
A
1
⁎
+
A
2
⁎
,

where
(3.3)
A
1
⁎
=
1
2
(
A
⁎
+
J
A
⁎
J
)
,
A
2
⁎
=
1
2
(
A
⁎
−
J
A
⁎
J
)
.

According to Lemma 1,
(3.4)
A
1
⁎
=
K
(
A
11
⁎
0
0
A
22
⁎
)
K
T
,

where
(3.5)
Q
1
T
A
11
⁎
P
1
=
(
A
111
⁎
A
112
⁎
A
113
⁎
A
121
⁎
A
122
⁎
A
123
⁎
A
131
⁎
A
132
⁎
A
133
⁎
)
,
Q
2
T
A
22
⁎
P
2
=
(
A
211
⁎
A
212
⁎
A
213
⁎
A
221
⁎
A
222
⁎
A
223
⁎
A
231
⁎
A
232
⁎
A
233
⁎
)
.

*Given X, Y, Λ, Γ according to* (2.5) *and A* _{0}. *If they satisfy the conditions of Theorem* 1*, and given A** ∈ *R* ^{ n×n } *, then Problem II has the unique solution*
*Moreover,*
*can be expressed as*

(3.6)
A
ˆ
=
A
10
+
E
F
ˆ
G
,

(3.7)
F
ˆ
=
K
(
F
ˆ
1
0
0
F
ˆ
2
)
K
T
,

(3.8)
F
ˆ
1
=
Q
1
(
A
011
A
012
Θ
3
−
1
A
113
⁎
Θ
1
−
1
A
021
Θ
1
−
1
(
A
022
−
Θ
2
A
222
⁎
Θ
4
)
Θ
3
−
1
A
123
⁎
A
131
⁎
A
132
⁎
A
133
⁎
)
P
1
T
,

(3.9)
F
ˆ
2
=
Q
2
(
A
211
⁎
A
212
⁎
A
213
⁎
A
221
⁎
A
222
⁎
Θ
2
−
1
A
023
A
231
⁎
A
032
Θ
4
−
1
A
033
)
P
2
T
.

Combining Theorem 1 and (3.2), for any *A* ∈ *S* _{ E }, we have

(3.10)
E
¯
+
E
=
I
n
,
E
¯
E
=
0
;
G
¯
+
G
=
I
n
,
G
¯
G
=
0
.

From this, we have
(3.11)
∥
E
A
1
⁎
G
−
E
F
ˆ
G
∥
=
0
.

Combining Lemma 1, (
2.11) and (
3.10), we have
Denote

(3.12)
(
A
011
A
012
Θ
3
−
1
F
113
Θ
1
−
1
A
021
Θ
1
−
1
(
A
022
−
Θ
2
F
222
Θ
4
)
Θ
3
−
1
F
123
F
131
F
132
F
133
)
=
(
A
111
⁎
A
112
⁎
A
113
⁎
A
121
⁎
A
122
⁎
A
123
⁎
A
131
⁎
A
132
⁎
A
133
⁎
)
+
(
F
¯
111
F
¯
112
F
¯
113
F
¯
121
F
¯
122
F
¯
123
F
¯
131
F
¯
132
F
¯
133
)
,

(3.13)
(
F
211
F
212
F
213
F
221
F
222
Θ
2
−
1
A
023
F
231
A
032
Θ
4
−
1
A
033
)
=
(
A
211
⁎
A
212
⁎
A
213
⁎
A
221
⁎
A
222
⁎
A
223
⁎
A
231
⁎
A
232
⁎
A
233
⁎
)
+
(
F
¯
211
F
¯
212
F
¯
213
F
¯
221
F
¯
222
F
¯
223
F
¯
231
F
¯
232
F
¯
233
)
.

(
3.12) and (
3.13) imply that if
(3.14)
F
¯
111
=
A
011
−
A
111
⁎
,
F
¯
112
=
A
012
Θ
3
−
1
−
A
112
⁎
,
F
¯
113
=
0
,
F
¯
121
=
Θ
1
−
1
A
021
−
A
121
⁎
,
F
¯
122
=
Θ
1
−
1
(
A
022
−
Θ
2
F
222
Θ
4
)
Θ
3
−
1
−
A
122
⁎
,
F
¯
123
=
0
,
F
¯
131
=
0
,
F
¯
132
=
0
,
F
¯
133
=
0
.
F
¯
211
=
0
,
F
¯
212
=
0
,
F
¯
213
=
0
,
F
¯
221
=
0
,
F
¯
222
=
F
222
−
A
222
⁎
,
F
¯
223
=
Θ
2
−
1
A
023
−
A
223
⁎
,
F
¯
231
=
0
,
F
¯
232
=
A
032
Θ
4
−
1
−
A
232
⁎
,
F
¯
233
=
A
033
−
A
233
⁎
.

According to (
3.14), we have
(3.15)
F
113
=
A
113
⁎
,
F
123
=
A
123
⁎
,
F
131
=
A
131
⁎
,
F
132
=
A
132
⁎
,
F
133
=
A
133
⁎
,
F
211
=
A
211
⁎
,
F
212
=
A
212
⁎
,
F
213
=
A
213
⁎
,
F
221
=
A
221
⁎
,
F
222
=
A
222
⁎
,
F
231
=
A
231
⁎
.

(
3.15) gives the results of Theorem 2.□
**Algorithm**

1. Input *A**, *A* _{0}, and input *X*, *Y*, *Λ*, *Γ* according to (2.5).

2. Compute

3. Compute *A* _{10}, *E*, *G* according to (2.9) and (2.10), and compute *A* _{110}, *A* _{210}, *E* _{1}, *E* _{2}, *G* _{1}, *G* _{2} according to (2.11).

4. Compute

5. Compute the GSVD of matrix pairs

6. Partition

7. Compute

8. Compute

9. Compute

10. Compute

11. Calculate

**Example** (*n* = 10, *h* = 6, *l* = 2, *p* = 3).

Give *J* and choose a random matrix *A* in GCSR^{10×10} as follows.

Compute the eigenvalues and the right eigenvectors of *A*, choose partial eigenpairs of *A* and obtain *X* _{1}, *X* _{2}, *Λ* _{1}, *Λ* _{2} according to (2.5).

In this article, we have obtained the necessary and sufficient conditions and associated general solutions of Problem I (Theorem 1). For given matrix *A** ∈ *R* ^{ n×n }, the unique optimal approximation solution of Problem II has been derived (Theorem 2). Our results extend and unify many results for left and right inverse eigenpairs problem, the inverse problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint, which is the first motivation of this work. For instance, in Problem I, if *Y* = 0, then this problem becomes Problem I in [17]; in Problem I, if *p* = 0, this problem becomes Problem I in [4,5,6,7,8,9,10,11,12,13].

The left and right eigenpairs of a real matrix are not all real eigenpairs, and its complex eigenpairs are all conjugate pairs. Hence, the supposition for Problem I in [4,5,6,7,10,11] is not suitable. In this article, we derive the suitable supposition for Problem I (*X*, *Y*, *Λ*, *Γ* are given by (2.5)), which is another motivation of this work.

This research was supported by the Natural Science Foundation of Hunan Province (2015JJ4090) and by Scientific Fund of Hunan Provincial Education Department of China (Grant no 13C1139). The authors are grateful to the anonymous reviewer and Dr. Justyna Zuk for their valuable comments and careful reading of the original manuscript of this article.

**Conflicts of interest:** The authors declare that there is no conflict of interests regarding the publication of this paper.

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