# The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

Authors: Jinrong Shen; Wei Liu; Baiyu Wang; Xiangyang Peng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.


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## Body

## 1. Introduction

Inverse eigenproblems arise in a remarkable variety of applications, including control theory [1, 2], vibration theory [3, 4], structural design [5], molecular spectroscopy [6], in developing numerical methods, and the ordinary and partial differential equation solving [7, 8]. Centrosymmetric matrices are applied in information theory, linear system theory, and numerical analysis theory [9]. The unconstrained centrosymmetric matrices’ problems have been discussed [9–14], a class of unconstrained matrices’ inverse eigenproblems has been obtained [15–18], and the constrained inverse eigenproblems have been discussed [19–22], but only when the eigenvalues are real or imaginary numbers. For general real matrices, the eigenvalues are not necessarily real or imaginary numbers, so when the eigenvalue is complex, it is difficult to find the constraint solution. In this paper, we will use the real Schur decomposition theorem and the similar decomposition theorem and introduce a new norm to get the corresponding expression of the best approximation solution.Throughout the paper, we denote the set of realn×m matrices, real n×n orthogonal matrices, n×n centrosymmetric matrices, and real numbers, respectively, by Rn×m, ORn×n, CSRn×n, and R. AT, A+, rankA, and AF denote the transpose, the Moore–Penrose generalized inverse, the rank, and the Frobenius norm of a matrix A, respectively. I is the identity matrix. λA denotes the set of eigenvalues of the matrix A. λA/λB denotes the set of the difference of λA and λB. Dr is a closed disc with radius r and center origin, ADr denotes the square matrix A with all of its eigenvalues located in the closed disc Dr, and RDrn×n denotes the set of n×n matrices with their eigenvalues located in the disc Dr. The notation A11⊕A22⊕⋯⊕Akk denotes the direct sum of the matrices A11,A22,…,Akk, where Ajj∈Rnj×nj. Let ei be the ith standard unit vector, and the matrix Sn=en,en−1,…,e1. x represents the largest integer less than or equal to x. The following definition is given in [23].Definition 1.
A
=
a
i
j
∈
R
n
×
n is called centrosymmetric matrix if aij=an+1−i,n+1−j, i,j=1,2,…,n.
Clearly, each eigenvalue of matrixA∈CSRn×n is either a real number or a complex number, if A has complex eigenvalues, and they must occur in complex conjugate pairs, note that λj=a˜j+ib˜j, λ¯j=a˜j−ib˜jb˜j≠0,j=1,2,…,k are eigenvalues of A, xj=ξj+ηji and x¯j=ξj−ηjij=1,2,…,k are eigenvectors associated with λj and λ¯j, respectively; also, we have(1)Aξjηj=ξjηja˜jb˜j−b˜ja˜j,j=1,2,…,k,where let(2)Xj=ξj,ηj,Λj=a˜jb˜j−b˜ja˜j.
IfA has real eigenvalues, then λj∈Rj=k+1,…,l are eigenvalues of A and xj∈Rnj=k+1,…,l are eigenvectors associated with λj; also, we have(3)Axj=λjxj,j=k+1,…,l,where let(4)Xj=xj,Λj=λj,
Then,2k+l−k=n.Remark 1.
Here,r-multiple eigenvalues are counted r-times and their eigenvectors may be linearly dependent.Definition 2.
LetA∈Rn×n, given Y∈Rn×n, and rankY=n. Then, we define a new norm called the Y norm of matrix A as(5)AY=Y−1AYF.
In contrast to the definition of matrix norm (see [24], Definition 5.1.1), it is easy to show that ⋅Y is a kind of matrix norm.
Now, we can present the optimal approximation of constrained inverse eigenproblem of centrosymmetric matrices as follows:Constrained Inverse Eigenproblem. Given a real number r>0. Let X=X1,X2,…,Xl∈Rn×m,Λ=diagΛ1,Λ2,…,Λl∈Rm×m,l≤m≤n, where XjandΛj satisfy equation (2) or equation (4). Dr=ZZ≤r,Z∈C is a given closed disc. Find matrix A such that the set(6)S=A∈CSRn×nAX=XΛ.is nonempty, and find the subsetSDr⊂S such that the remaining eigenvalues of any matrix in S are located in the disc Dr.Optimal Approximation Problem. Given A∗∈Rn×n, ∥⋅∥Y is the norm which has been defined as equation (5), Y is an invertible matrix concerning with A∗ and A∈SDr; find a matrix A^∈SDr, such that(7)A∗−A^Y=min∀A∈SDrA∗−AY,where SDr is the solution set of the Constrained Inverse Eigenproblem.
This paper is organized as follows. In Section2, we provide the solvability conditions of the Constrained Inverse Eigenproblem and its general solution in that case. In Section 3, we get the expression of the solution for Optimal Approximation Problem. In Section 4, we give an algorithm of Constrained Inverse Eigenproblem and Optimal Approximation Problem and give a numerical example of Optimal Approximation Problem.


## 2. The Solvability Conditions and General Solution of Constrained Inverse Eigenproblem

Firstly, letk=n/2 and characterize the set of all centrosymmetric matrices as follows.Whenn=2k, let(8)D=12IkIkSk−Sk.Whenn=2k+1, let(9)D=12Ik0Ik020Sk0−Sk.Clearly, D is an orthogonal matrix for all of the n.Secondly, givenX∈Rn×m, denote(10)DTX=XD1XD2,XD1∈Rn−k×m,XD2∈Rk×m.Decomposing the matricesX,XD1, and XD2 by the SVD, we have(11)X=U˜Δ000V˜T=U˜1ΔV˜1T,where U˜=U˜1,U˜2∈ORn×n,V˜=V˜1,V˜2∈ORm×m,r0=rankX, and(12)Δ=diagδ1,δ2,…,δr0,δi>0,1≤i≤r0,(13)XD1=UΓ000VT=U1ΓV1T,where U=U1,U2∈ORn−k×n−k,V=V1,V2∈ORm×m,r1=rankXD1, and Γ=diagγ1,γ2,…,γr1,γi>0,1≤i≤r1.(14)XD2=PΣ000QT=P1ΣQ1T,where P=P1,P2∈ORk×k,Q=Q1,Q2∈ORm×m,r2=rankXD2, and Σ=diagσ1,σ2,…,σr2,σi>0,1≤i≤r2.Lemma 1.
(see [9], Lemma 2).A∈CSRn×n if and only if A can be expressed as(15)A=DB00CDT,B∈Rn−k×n−k,C∈Rk×k,where, if n=2k, then D is the form of equation (8), and if n=2k+1, then D is the form of (9).Lemma 2.
(see [15], Theorems 7.2 and 7.3). Given X∈Rn×m,Λ∈Rm×m, and X decomposed as equation (11), let(16)U˜TAU˜=A11A12A21A22,where Aij=U˜iTAU˜ji,j=1,2. Equation AX=XΛ is consistent if and only if(17)XΛV˜2=0.
If the condition is satisfied, the general solution can be expressed as(18)A=XΛX++GU˜2T,and G=U˜A12A22∈Rn×n−r0 is arbitrary.Lemma 3.
GivenX∈Rn×mandΛ∈Rm×m. Equation AX=XΛ is consistent with A∈CSRn×n if and only if(19)XD1ΛV2=0,XD2ΛQ2=0,where XD1,XD2,V2,andQ2 is the same as equations (10)–(14). Let(20)UTBU=B11B12B21B22,PTCP=C11C12C21C22,where Bij=UiTBUjandCij=PiTCPji,j=1,2, and the set of the general solution can be expressed as(21)S=AA=DXD1ΛXD1++G1U2T00XD2ΛXD2++G2P2TDT,where G1=UB12B22∈Rn−k×n−k−r1andG2=PC12C22∈Rk×k−r2are arbitrary.Proof.
IfS is nonempty, by Lemma 1, we have(22)A=DB00CDT,B∈Rn−k×n−k,C∈Rk×k.
Using equation (10), equation AX=XΛ is equivalent to(23)BXD1=XD1Λ,CXD2=XD2Λ.
It follows from Lemma2, and we can obtain AX=XΛ is solvable if and only if equation (19) holds, and the solution set can be expressed as equation (21) easily.Remark 2.
Equation (21) implies that λA/λΛ∈SDr if and only if arbitrary matrices λB22∈SDr and λC22∈SDr. From Lemmas 1–3, we have the following theorem.Theorem 1.
Constrained Inverse Eigenproblem is solvable if and only if(24)XD1ΛV2=0,XD2ΛQ2=0.The general solution can be expressed as(25)A=DXD1ΛXD1++UB12B22Dr00XD2ΛXD2++PC12C22DrDT,where B12∈Rr1×n−k−r1, C12∈Rr2×k−r2, B22Dr∈RDrn−k−r1×n−k−r1, and C22Dr∈RDrk−r2×k−r2 are arbitrary matrices.


## 3. The Solution of Optimal Approximation Problem

Lemma 5.
Given a matrix(26)B˜jj=cjd1j−d2jcj,d1jd2j>0,λB˜jj2=cj2+d1jd2j.
By similarity invariant, we can derive an invertible matrixYB˜jj and YB˜jjF=1 such that(27)B˜jj=YB˜jjBjjYB˜jj−1,where Bjj=cjdj−djcj and λBjj2=cj2+dj2=λB˜jj2.Lemma 6.
GivenBjj=cjdj−djcjandλBjj=cj2+dj2=sj>r>0, then, there exists a unique matrix:(28)Ajj=ajbj−bjaj,aj2+bj2=r,such that(29)Ajj−BjjF=min,(30)Ajj=rλBjjBjj.Proof.
From(31)Ajj−BjjF2=min,we may obtain(32)aj−cj2+bj−dj2=min.
Letdj/cj=tanθj,bj/aj=tanθ,0≤θ,θj≤π,sj>r, then equation (32) is equivalent to(33)minfθ,r=minrcosθ−sjcosθj2+rsinθ−sjsinθj2=minr2+sj2−2rsjcosθ−θj.
It is easy to see thatfθ,r=min if and only if θ=θjandr=sj.
NoteBjjDr=Ajj, from equation (33), we may show that equation (29) implies that(34)BjjDr=Ajj=rsjBjj=rλBjjBjj.
There, we noteBjjDr=Ajj just for the sake of writing.
To show Theorem2, we first introduce some notations as follows:(35)DTA∗D=A11∗A12∗A21∗A22∗,where A11∗∈Rn−k×n−k,A12∗∈Rn−k×k,A21∗∈Rk×n−k,andA22∗∈Rk×k.Lemma 7.
(see [16], Theorem 7.6). If H∈Rn×n, then there exists an orthogonal matrix Q˜∈ORn×n, such that Q˜THQ˜ is upper quasi-triangular, that is, Q˜THQ˜ is block upper-triangular where each diagonal block is either a 1×1 matrix or a 2×2 matrix having complex conjugate eigenvalues.Lemma 8.
Given matrixA∗∈Rn×n, DTA∗D is decomposed as equation (35), the orthogonal matrices U and P are given in equations (13) and (14), respectively; then, there exist orthogonal matrices Q1∗andQ2∗ and inverse matrices Y1 and Y2, such that(36)U2TA11∗U2=Q1∗TY1Λ¯11∗⋯⋯⋯∗0⋱⋱⋮⋮⋱Λ¯ss⋱⋮⋮⋱λs+1⋱⋮⋮⋱⋱∗0⋯⋯⋯0λn−k−r1−2sY1−1Q1∗,where Λ¯jj=ljfj−fjlj,j=1,2,…,s.(37)P2TA22∗P2=Q2∗TY2Γ¯11∗⋯⋯⋯∗0⋱⋱⋮⋮⋱Γ¯tt⋱⋮⋮⋱μt+1⋱⋮⋮⋱⋱∗0⋯⋯⋯0μk−r2−2tY2−1Q2∗,where Γ¯jj=gjhj−hjgj,j=1,2,…,t.Proof.
For Lemma7, there exist orthogonal matrices Q1∗∈ORn−k−r1×n−k−r1andQ2∗∈ORk−r2×k−r2 such that(38)U2TA11∗U2=Q1∗TΛ11∗⋯⋯⋯∗0⋱⋱⋮⋮⋱Λss⋱⋮⋮⋱λs+1⋱⋮⋮⋱⋱∗0⋯⋯⋯0λn−k−r1−2sQ1∗,P2TA22∗P2=Q2∗TΓ11∗⋯⋯⋯∗0⋱⋱⋮⋮⋱Γtt⋱⋮⋮⋱μt+1⋱⋮⋮⋱⋱∗0⋯⋯⋯0μk−r2−2tQ2∗,where(39)Λjj=ljf1j−f2jlj,λj∈R,j=s+1,…,n−k−r1−2s,Γjj=gjh1j−h2jgj,μj∈R,j=t+1,…,k−r2−2t.For Lemma5, there exist inverse matrices Y1j and Y2j, satisfying(40)Y1j−1ΛjjY1j=ljfj−fjlj=Λ¯jj,Y1jF=1,j=1,2,…,s,Y2j−1ΓjjY2j=gjhj−hjgj=Γ¯jj,Y2jF=1,j=1,2,…,t.LetY1=Y11⊕⋯⊕Y1s⊕In−k−r1−2s and Y2=Y21⊕⋯⊕Y2t⊕Ik−r2−2t, In−k−r1−2s and Ik−r2−2t are identity matrices. Then, it is easy to see that equations (36) and (37) exist.Theorem 2.
GivenA∗∈Rn×n, suppose given matrices XandΛ, real number r>0 make Constrained Inverse Eigenproblem solvable, and the solution expression is the same as equation (25). Let(41)Y=DU00PIr10000Q1∗TY10000Ir20000Q2∗TY2.
Then, Optimal Approximation Problem has a unique solution which can be expressed as(42)A^=DB^00C^DT,where(43)B^=UΣ1V1TΛV1Σ1−1U1TA11∗U20B22DrUT,C^=PΣ2Q1TΛQ1Σ2−1P1TA22∗P20C22DrPT.Proof.
LetA∈SDr, by Definition 2, we have(44)A−A∗Y2=Y−1AY−Y−1A∗YF2=min∀A∈SDr.
The expression of A is the same as equation (25) and DTA∗D is decomposed as equation (35); then, equation (44) is equivalent to(45)Σ1V1TΛV1Σ1−1−U1TA11∗U12+B12Q1∗TY1−U1TA11∗U2Q1∗TY12+Y1−1Q1∗B22DrQ1∗TY1−Y1−1Q1∗U2TA11∗U2Q1∗Y12+Σ2Q1TΛQ1Σ2−1−P1TA22∗P12+C12Q12∗TY2−P1TA22∗P2Q2∗TY22+Y2−1Q2∗C22DrQ2∗TY2−Y2−1Q2∗P2TA22∗P2Q2∗TY22=min.
Obviously, we may derive that equation (44) has the solution A^ if and only if(46)Y1−1Q1∗U2TA11∗U2Q1∗TY1−Y1−1Q1∗B22DrQ1∗TYF2=min,(47)Y2−1Q2∗P2TA22∗P2Q2∗TY2−Y2−1Q2∗C22DrQ2∗Y2F2=min,(48)U1TA11∗U2Q1∗TY1−B12Q1∗TY1F=min,(49)P1TA22∗P2Q2∗TY2−C12Q2∗TY2F=min.
From Lemmas5 and 6, equations (36) and (37), we may see that equations (46) and (47) imply that(50)B22Dr=Q1∗TY1Λ¯11Dr∗⋯⋯⋯∗0⋱⋱⋮⋮⋱Λ¯ssDr⋱⋮⋮⋱λ¯s+1Dr⋱⋮⋮⋱⋱∗0⋯⋯⋯0λ¯n−k−r1−2sDrY1−1Q1∗,where(51)Λ¯jjDr=rλΛ¯jjΛ¯jj,λΛ¯jj>r,Λ¯jj,λΛ¯jj≤r,λ¯jjDr=signλ¯ir,λi>r,λi,λi≤r,j=1,2,…,s,i=s+1,…,n−k−r1−2s.(52)C22Dr=Q2∗TY2Γ¯11Dr∗⋯⋯⋯∗0⋱⋱⋮⋮⋱Γ¯ttDr⋱⋮⋮⋱μ¯t+1Dr⋱⋮⋮⋱⋱∗0⋯⋯⋯0μ¯k−r2−2tDrY2−1Q2∗,where(53)Γ¯jjDr=rλΓ¯jjΓ¯jj,λΓ¯jj>r,Γ¯jj,λΓ¯jj≤r,μ¯jjDr=signμ¯ir,μi>r,μ¯i,μi≤r,j=1,…,t,i=t+1,…,k−r2−2t.
By equations (48) and (49), we have(54)B12=U1TA11∗U2,C12=P1TA22∗P2.
Note(55)B^=UΣ1V1TΛV1Σ1−1U1TA11∗U20B22DrUT,(56)C^=PΣ2Q1TΛQ1Σ2−1P1TA22∗P20C22DrPT.
Equations (55), (56), and (25) imply equation (42).


## 4. Numerical Example

Based on Theorems1 and 2, we propose the following algorithm for solving Constrained Inverse Eigenproblem and Optimal Approximation Problem (Algorithm 1).Algorithm 1: The solution of (7).
Step 1. InputX,Λ,A∗,r;
Step 2. DecomposeDTX as equation (10);
Step 3. Calculate the singular value decompositionXD1,XD2 as equations (13) and (14), verify that equation (24) is true and A can be expressed as equation (25);
Step 4. DecomposeDTA∗D as equation (35);
Step 5. DecomposeU2TA11∗U2,P2TA22∗P2 as equations (36), (37), respectively;
Step 6. CalculateB22Dr,C22Dr as equations (50) and (52), respectively;
Step 7. CalculateA^ by Theorem 2.In this section, we will give a numerical example to illustrate our results. All the tests are performed by MATLAB6.5.The matricesX, Λ, and A∗ and the radius r are given by following:(57)X=−0.0638−0.1385−0.0249−0.0593−0.3027−0.0061−0.33010.05210.03470.61920.62020.00000.6192−0.62020.0000−0.3301−0.0521−0.0347−0.05930.30270.0061−0.06380.13850.0249,Λ=0.76110.00000.00000.00000.55290.07020.0000−0.07020.5529,A∗=0.95010.82140.93550.13890.44510.83810.30460.37840.23110.44470.91690.20280.93180.01960.18970.86000.60680.61540.41030.19870.46600.68130.19340.85370.48600.79190.89360.60380.41860.37950.68220.59360.89130.92180.05790.27220.84620.83180.30280.49660.76210.73820.35290.19880.52520.50280.54170.89980.45650.17630.81320.01530.20260.70950.15090.82160.01850.40570.00990.74680.67210.42890.69790.6449,r=0.4.By algorithm, the constrained optimal approximation solutionA^ by Optimal Approximation Problem is(58)A^=0.45390.2884−0.17280.0469−0.10800.07820.0071−0.07100.2569−0.0060−0.1473−0.0179−0.04760.2079−0.3727−0.14640.14500.1242−0.0729−0.2928−0.2391−0.35260.02050.69770.08320.25220.35220.54420.2411−0.54500.39830.54500.54500.3983−0.54500.24110.54420.35220.25220.08320.69770.0205−0.3526−0.2391−0.2928−0.07290.12420.1450−0.1464−0.37270.2079−0.0476−0.0179−0.1473−0.00600.2569−0.07100.00710.0782−0.10800.0469−0.17280.28840.4539.In the practical engineering problems, it is usually required that the matrix is a centrosymmetric matrix and the partial eigenvalues of the matrix must be located in a given closed disc or interval. We verify thatA^ in Optimal Approximation Problem is a centrosymmetric matrix and the eigenvalues of λA^/λΛ are lost in the disc Dr.



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