We discuss the feasible interval of the parameter k and a general expression of matrix X which satisfies the rank equation r(A-BXC)=k. With these results, we study two problems under the rank constraint r(A-BXC)=k. The first one is to determine the maximal and minimal ranks under the rank constraint r(A-BXC)=k. The second one is to derive the least squares solutions of ∥BXC-A∥F=min under the rank constraint r(A-BXC)=k.

1. Introduction

We adopt the following notation in this paper. The set of m×n matrices with complex entries is denoted by Cm×n. The conjugate transpose of a matrix A is denoted by A*. The symbols Ik and r(A) are the k×k identity matrix and the rank of A∈Cm×n, respectively. ∥·∥ stands for the matrix Frobenius norm. The Moore-Penrose inverse of A∈Cm×n is defined as the unique matrix X∈Cn×m satisfying
(1)(1)AXA=A,(2)XAX=X,(3)(AX)*=AX,(4)(XA)*=XA,
and is denoted by X=A† (see [1]). Furthermore, we denote EA=Im-AA† and FA=In-A†A.

In the literature, ranks of solutions of linear matrix equations have been studied widely. Uhlig [2] derived the extremal ranks of solutions of the consistent matrix equation of AX=B. Tian [3] derived the extremal ranks of solutions of BXC=A. Li and Liu [4] studied the extremal ranks of Hermitian solutions of AX=B. Li et al. [5] studied the extremal ranks of solutions with special structure of AX=B. Liu [6] derived the extremal ranks of solutions of AX+YB=C. Wang and Li [7] established the maximal and minimal ranks of the solution to consistent system A1X1=C1,A2X2=C2, and A3X1B1+A4X2B2=C3. Wang and He [8] derived the extremal ranks of the general solution of the mixed Sylvester matrix equations
(2)A1X-YB1=C1,A2Z-YB2=C2.
Liu [9] derived the extremal ranks of least square solutions to BXC=A. Sou and Rantzer [10] studied the minimum rank matrix approximation broblem in the spectral norm
(3)minXrank(X)subjectto∥A-BXC∥2<1.
Wei and Shen [11] studied a more general problem
(4)minXrank(X)subjectto∥A-BXC∥2<ξ,
where ξ≥θ and θ=minY∥A-BYC∥2. More results and applications about ranks of matrix expressions and solutions of matrix equations can be seen in ([2, 3, 8, 11–13], etc.).

Motivated by the work of [2, 3, 7–9, 14, 15], we consider a general problem. Assume that k is a prescribed nonnegative integer and A∈Cm×n, B∈Cm×p, and C∈Cq×n are given matrices. We now investigate the problem to determine the maximal and minimal ranks of solutions to the rank equation r(A-BXC)=k. This problem can be stated as follows.

Problem 1.

Given matrices A∈Cm×n, B∈Cm×p, and C∈Cq×n and nonnegative integer k1, characterize the set
(5)Sk1={X∣X∈Cp×q,r(A-BXC)=k1},
and determine the maximal and minimal ranks of solutions of the rank equation r(A-BXC)=k1.

In [16–18], Wang, Wei, and Zha studied least squares solutions of line matrix equations under rank constraints, respectively. In [19], Wei and Wang derived a rank-k Hermitian nonnegative definite least squares solution to the equation BXB*=A. In Problem 2, we discuss the least squares solutions X of ∥BXC-A∥F=min subject to r(A-BXC)=k. This problem can be stated as follows.

Problem 2.

Given matrices A∈Cm×n, B∈Cm×p, and C∈Cq×n and nonnegative integer k, determine the range of k, such that there exists a least squares solution X of ∥BXC-A∥F=min subject to r(A-BXC)=k; that is, characterize the set
(6)S^={X∣X∈Cp×q,kkkkk∥A-BXC∥=minsubjecttor(A-BXC)=k}.

The paper is organized as follows. In Section 2, we provide some preliminary results; in Sections 3 and 4, we study Problems 1 and 2, respectively; and finally in Section 5, we conclude the paper with some remarks.

2. Preliminaries

In this section we present some preliminary results which will be used in the following sections to study Problems 1 and 2.

Lemma 3 (see [<xref ref-type="bibr" rid="B12">20</xref>]).

Let A∈Cm×n, B∈Cm×k, C∈Cl×n, and D∈Cl×k be given. Then
(7)r[AB]=r(A)+r(EAB)=r(B)+r(EBA),(8)r[AC]=r(A)+r(CFA)=r(C)+r(AFC),(9)r[ABC0]=r(B)+r(C)+r(EBAFC),
where B1=EAB and C1=CFA.

Lemma 4.

Let A∈Cm×n be given. Then
(10)minr(X)=kr(A-X)=max{k-r(A),r(A)-k},(11)maxr(X)=kr(A-X)=min{m,n,k+r(A)}.

Lemma 5 (see [<xref ref-type="bibr" rid="B14">21</xref>]).

Let A∈Cm×n, B∈Cm×p, C∈Cq×n, and D∈Cq×p be given. Then
(12)r(D-CA†B)=r[A*AA*A*BCA*D]-r(A).

Lemma 6 (see [<xref ref-type="bibr" rid="B4">22</xref>, <xref ref-type="bibr" rid="B11">23</xref>] (the Eckart-Young-Mirsky theorem)).

Let A∈Crm×n, k be a given nonnegative integer in which k≤r and the singular value decomposition [24] of A be
(13)A=U[Σ000]V*,
where Σ=diag{λ1,…,λr},λ1≥⋯≥λr>0, and U and V are unitary matrices of appropriate sizes. Then
(14)minr(X)=k∥A-X∥2=∑i=k+1rλi2.
Furthermore, when λk>λk+1,
(15)X=Udiag{λ1,…,λk,0,…,0}V*;
when p<k<q≤r and λp>λp+1=⋯=λq>λq+1,
(16)X=Udiag{λ1,…,λp,λkQQ*,0,…,0}V*,
where Q is an arbitrary matrix satisfying Q∈C(q-p)×(k-p) and Q*Q=Ik-p.

3. Solutions to Problem <xref ref-type="statement" rid="problem1.1">1</xref>

In this section, we study Problem 1 proposed in Section 1.

Suppose that the matrices A∈Cm×n, B∈Cm×p, and C∈Cq×n are given. Let
(17)l=r[AC]+r[AB]-r[ABC0].
Then from [25] there exists X such that r(A-BXC)=k1, if and only if
(18)min{r[AB],r[AC]}≥k1≥l.
Furthermore, let
(19)B=U1[Σ1000]V1*,C=U2[Σ2000]V2*
be singular value decompositions of B and C with unitary matrices U1∈Cm×m, U2∈Cq×q, V1∈Cp×p, and V2∈Cn×n. Write U1*AV2 in partitioned form as
(20)U1*AV2=[A11A12A21A22],
where A11∈Cr(B)×r(C), A12∈Cr(B)×(n-r(C)), A21∈C(m-r(B))×r(C), and A22∈C(m-r(B))×(n-r(C)). Also assume that the singular value decomposition of A22 and the corresponding decompositions are given by
(21)A22=U3[Σ3000]V3*,U3*A21=[JK],A12V3=[EF],F=U4[Σ4000]V4*,K=U5[Σ5000]V5*,U4*(A11-EΣ3-1J)V5=[A^11A^12A^21A^22],
where Ui and Vi are unitary matrices of appropriate sizes in which i=3,4,5, J∈Cr(A22)×r(C), K∈C(m-r(B)-r(A22))×r(C), E∈Cr(B)×r(A22), F∈Cr(B)×(n-r(C)-r(A22)), A^11∈Cr(F)×r(K), A^12∈Cr(F)×(r(C)-r(K)), A^21∈C(r(B)-r(F))×r(K), and A^22∈C(r(B)-r(F))×(r(C)-r(K)).

We have the following result.

Theorem 7.

Suppose that the singular value decompositions of matrices B, C, A22, F, and K are given in (19)–(21). U1*AV2, U3*A21, U3*A21, A12V3, and U4*(A11-EΣ3-1J)V5 have the forms in (20) and (21). If k satisfies (18), then any solution X to the rank equation r(A-BXC)=k1 has the form(22)X=V1[Σ1-1U400Ip-r(B)][[Y11Y12Y21A^22+T]X12X21X22]×[V5*Σ2-100Iq-r(C)]U2*,
where X12∈Cr(B)×(q-r(C)), X21∈C(p-r(B))×r(C), X22∈C(p-r(B))×(q-r(C)), Y11∈Cr(F)×r(K), Y12∈Cr(F)×(r(C)-r(K)), and Y21∈C(r(B)-r(F))×r(K) are arbitrary, T∈C(r(B)-r(F))×(r(C)-r(K)), and r(T)=k1-l.

Proof.

From the singular value decompositions of matrices of B,C,F, and K, we observe that
(23)U1[A11000]V2*=BB†AC†C,U1[0A1200]V2*=BB†AFC,U1[00A210]V2*=EBAC†C,U1[000A22]V2*=EBAFC,U1[0A120A22]V2*=AFC,U1[00A21A22]V2*=EBA,[0F]V3*=A12FA22,U3[0K]=EA22A21,U4[000A^22]V5*=EF(A11-EΣ3-1J)FK.
Then by repeated application of Lemma 3, we have
(24)r(A22)=r(EBAFC)=r[ABC0]-r(B)-r(C),(25)r(F)=r([0F]V3*)=r(A12FA22)=r[A12A22]-r(A22)=r(AFC)-r(EBAFC)=r(B)+r[AC]-r[ABC0],(26)r(K)=r(U3[0K])=r(EA22A21)=r[A21A22]-r(A22)=r(EBA)-r(EBAFC)=r(C)+r[AB]-r[ABC0].
Furthermore, write
(27)V1*XU2=[Σ1-1X11Σ2-1X12X21X22],U4*X11V5=[Y11Y12Y21Y22]
in which X11∈Cr(B)×r(C), X12∈Cr(B)×(q-r(C)), X21∈C(p-r(B))×r(C), X22∈C(p-r(B))×(q-r(C)), Y11∈Cr(F)×r(K), Y12∈Cr(F)×(r(C)-r(K)), Y21∈C(r(B)-r(F))×r(K), and Y22∈C(r(B)-r(F))×(r(C)-r(K)). It follows that(28a)r(A-BXC)=r(U1*AV2-[Σ1000]V1*XU2[Σ2000])=r[A11-X11A12A21A22](28b)=r[A11-X11A12V3U3*A21[Σ3000]]=r[A11-X11[EF][JK][Σ3000]]=r[A11-EΣ3-1J-X11FK0]+r(A22)(28c)=r[[A^11A^12A^21A^22]-U4*X11V5[Σ4000][Σ5000]0]+r(A22)=r[A^11-Y11A^12-Y12Σ40A^21-Y21A^22-Y2200Σ50000000]+r(A22)=r(A^22-Y22)+r(A22)+r(F)+r(K).

Since r(A-BXC)=k1, from (28c), we obtain
(29)r(A^22-Y22)=k1-r(A22)-r(F)-r(K).

The identity r(A^22-Y22)=k1-l follows by substituting (24)–(26) into (29). Hence, any solution Y22 to the rank equation r(A^22-Y22)=k1-l has the form
(30)Y22=A^22+T,
where r(T)=k1-l.

Substituting (30) into the second partitioned matrix in (27), we obtain
(31)X11=U4[Y11Y12Y21A^22+T]V5*,
where r(T)=k1-l. The expression of X in (22) follows by substituting (31) into the first partitioned matrix in (27).

Let X=0. From (28b), we have
(32)r[A11-EΣ3-1JFK0]=r(A)-r(A22).
Substituting (24) into the above identity, we have
(33)r[A11-EΣ3-1JFK0]=r(A)+r(B)+r(C)-r[ABC0].
By applying Lemma 3 (9) to the final identity in (23), it follows that
(34)r(A^22)=r(EF(A11-EΣ3-1J)FK)=r[A11-EΣ3-1JFK0]-r(F)-r(K)=r(A)+r[ABC0]-r[AC]-r[AB]=r(A)-l.

We have the following result.

Theorem 8.

Let A, B, and C be as in Problem 1 and let k1 satisfy (18). Then
(35)minr(A-BXC)=k1r(X)=max{r(A)-k1,k1-r(A)},(36)maxr(A-BXC)=k1r(X)=min{-2r(B)-2r(C)+r[AC]+r[AB]p,q,k1+p+q+r(A)kkkkkkkkk-2r(B)-2r(C)+r[AC]+r[AB]}.

Proof.

From a general expression of X for the rank equation r(A-BXC)=k1 given in (22), (10), and (34), we obtain
(37)minr(A-BXC)=k1r(X)=minr(T)=k1-lr(A^22+T)=max{k1-l-r(A^22),r(A^22)-k1+l}=max{r(A)-k1,k1-r(A)}.

From (11), (22), and (34), we obtain
(38)maxr(A-BXC)=k1r(X)=maxr(T)=k1-l,Y11,Y12,Y21,X12,X21,X22r[[Y11Y12Y21A^22+T]X12X21X22]=maxr(T)=k-l,Y∈Cp×q,Z∈Cp×qr([A^22+T000]-Y[000Iq-r(C)+r(K)]kkkkkkkkkkkkkkkkkkkk-[000Ip-r(B)+r(F)]Z)=min{maxr(T)=k-lr[A22^+T000000Ip-r(B)+r(F)00000Iq-r(C)+r(K)00]p,q,p+q,kkkkkkmaxr(T)=k-lr[A^22+T000000Ip-r(B)+r(F)00000Iq-r(C)+r(K)00]}=min{p,q,maxr(T)=k1-lr(A^22+T)+p+qkkkkkkk+r(F)+r(K)-r(B)-r(C)p,q,maxr(T)=k1-lr(A^22+T)+p+q}=min{r[AC]+r[AB]p,q,p+q+r(K)-r(C),kkkkkkkkp+q+r(F)-r(B),kkkkkkkkk1+p+q+r(A)-2r(B)-2r(C)kkkkkkk+r[AC]+r[AB]}.
Since B∈Cm×p and C∈Cq×n, we see that p+q+r(K)-r(C)≥p and p+q+r(F)-r(B)≥q. To simplify expression (38) by the two inequalities, we obtain expression (36) for the maximal rank of solutions to the rank equation r(A-BXC)=k1.

Remark 9 (see [<xref ref-type="bibr" rid="B15">3</xref>]).

Let A, B, and C be as in Theorem 7. The matrix equation BXC=A is consistent, if and only if there exists X such that r(A-BXC)=0. Therefore, applying Theorem 8, we have the extremal ranks of solutions to the matrix equation BXC=A:
(39)minBXC=Ar(X)=r(A),maxBXC=Ar(X)=min{p,q,p+q+r(A)-r(B)-r(C)}.

Remark 10 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let A, B, and C be as in Theorem 7 and let S={X∈Cp×q∣∥BXC-A∥=min}. Since X∈S, if and only if B*AC*=B*BXCC*, and the matrix equation B*AC*=B*BXCC* is always consistent, we can use B*AC*, B*B, and CC* to replace A, B, and C in (39). Then we have the extremal ranks of least squares solutions of the matrix equation BXC=A:
(40)min∥BXC-A∥=minr(X)=r(B*AC*),max∥BXC-A∥=minr(X)=min{p,q,p+qkkkkkkk+r(B*AC*)-r(B)-r(C)}.

In [14], Liu and Tian derive the extremal ranks of submatrices in a Hermitian solution to the consistent matrix equation BXB*=A. In the following theorem, we derive the range of k1 such that there exists a Hermitian solution X to the rank equation r(A-BXB*)=k1, and the maximal and minimal ranks of X which may be proved in the same way as Theorem 8.

Theorem 11.

Let A∈Cm×m and B∈Cm×n be given, and let A be Hermitian. Then from [15] there exists a Hermitian matrix X satisfying r(A-BXB*)=k1, if and only if
(41)r[AB]≥k1≥2r[AB]-r[ABB*0].
If k1 satisfies the above inequalities, then
(42)minr(A-BXB*)=k1r(X)=max{r(A)-k1,k1-r(A)},maxr(A-BXB*)=k1r(X)=min{n,k1+2n+r(A)kkkkkk-4r(B)+2r[AB]}.

4. Solutions to Problem <xref ref-type="statement" rid="problem1.2">2</xref>

In this section, we study Problem 2 proposed in Section 1.

Let
(43)A=[1000],B=C=I2.
It is obvious that
(44)2≥r(A-BXC)≥0
and there do not exist the least squares solutions of ∥BXC-A∥F=min subject to r(A-BXC)=2. Therefore, we should study the range of k, such that there exists a least squares solution X of ∥BXC-A∥F=min subject to r(A-BXC)=k.

Theorem 12.

Let A, B, and C be as in Theorem 7. Then there exists a least squares solution of ∥A-BXC∥=min under the rank constraint r(A-BXC)=k, if and only if
(45)r[AAC*BB*A00C00]-r(B)-r(C)≥k≥r[AC]+r[AB]-r[ABC0].

Proof.

Let E, F, J, K, Σi, Ui, and Vi(i=1,2,3,4,5) be as in Theorem 7, and let U4*(EΣ3-1J)V5 be partitioned in the form
(46)U4*(EΣ3-1J)V5=[D11D12D21D22],
where D22∈C(r(B)-r(F))×(r(C)-r(K)), D21∈C(r(B)-r(F))×r(K), D12∈Cr(F)×(r(C)-r(K)), and D11∈Cr(F)×r(K). Let D22 have the singular value decomposition
(47)D22=U7[Σ7000]V7*,
where Σ7=diag{δ1,…,δr(D22)},δ1≥⋯≥δr(D22)>0, and U7 and V7 are unitary matrices of appropriate sizes.

From the partitioned form for U4*(A11-EΣ3-1J)V5 in (21),
(48)U4*A11V5=U4*(EΣ3-1J)V5+[A^11A^12A^21A^22].
Since the Frobenius norm is invariant, we have the following identities by substituting (22) into ∥A-BXC∥ and applying (46) and (48):
(49)minr(A-BXC)=k∥A-BXC∥2=minY11,Y12,Y21,r(T)=k-l∥[A11-U4[Y11Y12Y21A^22+T]V5*A12A21A22]∥2=∥[0A12A21A22]∥2+minY11,Y12,Y21,r(T)=k-l∥U4*A11V5-[Y11Y12Y21A^22+T]∥2=∥[0A12A21A22]∥2+minY11,Y12,Y21,r(T)=k-l∥U4*(EΣ3-1J)V5-[Y11Y12Y21T]∥2=∥[0A12A21A22]∥2+minr(T)=k-l∥D22-T∥2.
Therefore, there exists a least squares solution X satisfying ∥A-BXC∥2=min subject to r(A-BXC)=k if and only if r(D22)≥r(T)≥0, that is, if and only if
(50)r(D22)+l≥k≥l.

From the partitioned form for U4*(EΣ3-1J)V5 in (46) and the decompositions of F and K in (21), we have r(D22)=r(EFEΣ3-1JFK). Applying (9) gives
(51)r(D22)=r[EΣ3-1JFK0]-r(F)-r(K).
The identity
(52)r(A-BB†AC†C)=r[0A12A21A22]
follows from applying the decompositions of B and C in (19) and the partitioned form for U1*AV2 in (20). Substituting the decomposition of A22 in (19) into [0A12A21A22], applying the partitioned forms for U3*A21 and A12V3 in (21), we conclude that
(53)r(A-BB†AC†C)=r[EΣ3-1JFK0]+r(A22).
Hence,
(54)r(D22)=r(A-BB†AC†C)-r(A22)-r(F)-r(K).
It follows from applying (12) that
(55)r(A-BB†AC†C)=r[AAC*BB*A00C00]-r(B)-r(C).

Substituting (24)–(26) and (55) into (54), we have
(56)r(D22)=r[AAC*BB*A00C00]+r[ABC0]-r[AC]-r[AB]-r(B)-r(C).
Therefor, the inequalities in (45) follow from substituting (17) and (56) into (50).

Theorem 13.

Let A, B, C, E, F, J, K, Σi, Ui, and Vi(i=1,2,3,4,5) be as in Theorem 7, and let U4*(EΣ3-1J)V5 and D22 be partitioned as in (46) and (47), respectively. If k satisfies (45), then any least squares solution X∈{X∣r(A-BXC)=k} satisfying ∥A-BXC∥=min has the form
(57)X=V1[Σ1-1U400Ip-r(B)][[Y11Y12Y21A^22+T]X12X21X22]×[V5*Σ2-100Iq-r(C)]U2*,
where X12∈Cr(B)×(q-r(C)), X21∈C(p-r(B))×r(C), X22∈C(p-r(B))×(q-r(C)), Y11∈Cr(F)×r(K), Y12∈Cr(F)×(r(C)-r(K)), Y21∈C(r(B)-r(F))×r(K), and T∈C(r(B)-r(F))×(r(C)-r(K)) are arbitrary matrices, such that r(T)=k-l.

When δk-l>δk-l+1,
(58)T=U7diag{δ1,…,δk-l,0,…,0}V7*;

when p^<k-l<q^≤r and λp^>λp^+1=⋯=δq^>δq^+1,
(59)T=U7diag{δ1,…,δp^,δk-lQQ*,0,…,0}V7*,

where Q is an arbitrary matrix satisfying Q∈C(q^-p^)×(k-l-p^) and Q*Q=Ik-l-p^.
Proof.

When k satisfies the inequalities in (45), then, by applying Lemma 6 and (48), we obtain the desired form of T in (58) and (59), respectively.

5. Conclusions

In this paper, we have discussed the solutions to Problem 1(60)Sk1={X∣X∈Cp×q,r(A-BXC)=k1}
and the solutions to Problem 2(61)S^={X∣X∈Sk,∥A-BXC∥2=min}.

We first derived the expression of solutions to r(A-BXC)=k1 when Problem 1 is solvable. Based on these results, we obtained the extremal ranks of the expression of solutions to Problem 1, the solvability conditions of Problem 2, and the expression of least squares solutions when Problem 2 is solvable.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. The work of the first author was supported in part by the National Natural Science Foundation of China (Grant no. 11171226). The work of the second author was supported in part by the University Natural Science Foundation of Anhui Province (Grant no. KJ2013A239) and the National Natural Science Foundation of China (Grant no. 11301529).

Ben-IsraelA.GrevilleT. N. E.UhligF.On the matrix equation AX=B with applications to the generators of a controllability matrixTianY.Ranks of solutions of the matrix equation AXB=CLiR.LiuY.Ranks of Hermitian solutions of the matrix equation AX=BLiY.ZhangF.GuoW.ZhaoJ.Solutions with special structure to the linear matrix equation AX=BLiuY. H.Ranks of solutions of the linear matrix equation AX+YB=CWangQ.-W.LiC.-K.Ranks and the least-norm of the general solution to a system of quaternion matrix equationsWangQ.HeZ. H.Solvability conditions and general solution for mixed Sylvester equationsLiuY. H.Ranks of least squares solutions of the matrix equation AXB=CSouK. C.RantzerA.On a generalized matrix approximation problem in the spectral normWeiM.ShenD.Minimum rank solutions to the matrix approximation problems in the spectral normDuanX. F.WangQ. W.LiJ. F.On the low-rank approximation arising in the generalized Karhunen-Loeve transformWangH.XuJ.Some results on characterizations of matrix partial orderingsLiuY.TianY.Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA*=B with applicationsTianY.LiuY.Extremal ranks of some symmetric matrix expressions with applicationsWangH.On least squares solutions subject to a rank restrictionWeiM.Perturbation theory for the Eckart-Young-Mirsky theorem and the constrained total least squares problemZhaH. Y.The restricted singular value decomposition of matrix tripletsWeiM.WangQ.On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXAH=BMarsagliaG.StyanG. P. H.Equalities and inequalities for ranks of matricesTianY.More on maximal and minimal ranks of Schur complements with applicationsEckartC.YoungG.The approximation of one matrix by another of lower rankMirskyL.Symmetric gauge functions and unitarily invariant normsGolubG. H.van LoanC. F.ChuD. L.ChanH. C.HoD. W. C.Regularization of singular systems by derivative and proportional output feedback