Citation: Øyvind Blaker, Roselyn Carroll, Priscilla Paniagua, Don J. DeGroot, Jean-Sebastien L'Heureux. Halden research site: geotechnical characterization of a post glacial silt[J]. AIMS Geosciences, 2019, 5(2): 184-234. doi: 10.3934/geosci.2019.2.184
[1] | Mahmoud S. Mehany, Faizah D. Alanazi . An η-Hermitian solution to a two-sided matrix equation and a system of matrix equations over the skew-field of quaternions. AIMS Mathematics, 2025, 10(4): 7684-7705. doi: 10.3934/math.2025352 |
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[8] | Anli Wei, Ying Li, Wenxv Ding, Jianli Zhao . Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation. AIMS Mathematics, 2022, 7(4): 5029-5048. doi: 10.3934/math.2022280 |
[9] | Dong Wang, Ying Li, Wenxv Ding . The least squares Bisymmetric solution of quaternion matrix equation AXB=C. AIMS Mathematics, 2021, 6(12): 13247-13257. doi: 10.3934/math.2021766 |
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In this paper, we establish the following four symmetric quaternion matrix systems:
{A11X1=B11,C11X1D11=E11,X2A22=B22,C22X2D22=E22,F11X1H11+X2F22=G11, | (1.1) |
{A11X1=B11,C11X1D11=E11,X2A22=B22,C22X2D22=E22,F11X1+H11X2F22=G11, | (1.2) |
{A11X1=B11,C11X1D11=E11,A22X2=B22,C22X2D22=E22,F11X1+H11X2F22=G11, | (1.3) |
{A11X1=B11,C11X1D11=E11,A22X2=B22,C22X2D22=E22,F11X1+X2F22=G11, | (1.4) |
where Aii, Bii, Cii, Dii, Eii, Fii(i=¯1,2), H11, and G11 are known matrices, while Xi(i=¯1,2) are unknown.
In this paper, R and Hm×n denote the real number field and the set of all quaternion matrices of order m×n, respectively.
H={v0+v1i+v2j+v3k|i2=j2=k2=ijk=−1,v0,v1,v2,v3∈R}. |
Moreover, r(A), 0 and I represent the rank of matrix A, the zero matrix of suitable size, and the identity matrix of suitable size, respectively. The conjugate transpose of A is A∗. For any matrix A, if there exists a unique solution X such that
AXA=A,XAX=X,(AX)∗=AX,(XA)∗=XA, |
then X is called the Moore-Penrose (M−P) inverse. It should be noted that A† is used to represent the M−P inverse of A. Additionally, LA=I−A†A and RA=I−AA† denote two projectors toward A.
H is known to be an associative noncommutative division algebra over R with extensive applications in computer science, orbital mechanics, signal and color image processing, control theory, and so on (see [1,2,3,4]).
Matrix equations, significant in the domains of descriptor systems control theory [5], nerve networks [6], back feed [7], and graph theory [8], are one of the key research topics in mathematics.
The study of matrix equations in H has garnered the attention of various researchers; consequently they have been analyzed by many studies (see, e.g., [9,10,11,12]). Among these the system of symmetric matrix equations is a crucial research object. For instance, Mahmoud and Wang [13] established some necessary and sufficient conditions for the three symmetric matrix systems in terms of M−P inverses and rank equalities:
{A1V=C1, VB1=C2,A3X+YB3=C3,A2Y+ZB2+A5VB5=C5,A4W+ZB4=C4,{A1V=C1, VB1=C2,A3X+YB3=C3,A2Z+YB2+A5VB5=C5,A4Z+WB4=C4,{A1V=C1, VB1=C2,A3X+YB3=C3,A2Y+ZB2+A5VB5=C5,A4Z+WB4=C4. | (1.5) |
Wang and He [14] established the sufficient and necessary conditions for the existence of solutions to the following three symmetric coupled matrix equations and the expressions for their general solutions:
{A1X+YB1=C1,A2Y+ZB2=C2,A3W+ZB3=C3,{A1X+YB1=C1,A2Z+YB2=C2,A3Z+WB3=C3,{A1X+YB1=C1,A2Y+ZB2=C2,A3Z+WB3=C3. | (1.6) |
It is noteworthy that the following matrix equation plays an important role in the analysis of the solvability conditions of systems (1.1)–(1.4):
A1U+VB1+A2XB2+A3YB3+A4ZB4=B. | (1.7) |
Liu et al. [15] derived some necessary and sufficient conditions to solve the quaternion matrix equation (1.7) using the ranks of coefficient matrices and M−P inverses. Wang et al. [16] derived the following quaternion equations after obtaining some solvability conditions for the quaternion equation presented in Eq (1.8) in terms of M−P inverses:
{A11X1=B11,C11X1D11=E11,X2A22=B22,C22X2D22=E22,F11X1+X2F22=G11. | (1.8) |
To our knowledge, so far, there has been little information on the solvability conditions and an expression of the general solution to systems (1.1)–(1.4).
In mathematical research and applications, the concept of η-Hermitian matrices has gained significant attention [17]. An η-Hermitian matrix, for η∈{i,j,k}, is defined as a matrix A such that A=Aη∗, where Aη∗=−ηA∗η. These matrices have found applications in various fields including linear modeling and the statistics of random signals [1,17]. As an application of (1.1), this paper establishes some necessary and sufficient conditions for the following matrix equation:
{A11X1=B11,C11X1Cη∗11=E11,F11X1Fη∗11+(F22X1)η∗=G11 | (1.9) |
to be solvable.
Motivated by the study of Systems (1.8), symmetric matrix equations, η-Hermitian matrices, and the widespread use of matrix equations and quaternions as well as the need for their theoretical advancements, we examine the solvability conditions of the quaternion systems presented in systems (1.1)–(1.4) by utilizing the rank equalities and the M−P inverses of coefficient matrices. We then obtain the general solutions for the solvable quaternion equations in systems (1.1)–(1.4). As an application of (1.1), we utilize the M−P inverse and the rank equality of matrices to investigate the necessary and sufficient conditions for the solvability of quaternion matrix equations involving η-Hermicity matrices. It is evident that System (1.8) is a specific instance of System (1.1).
The remainder of this article is structured as follows. Section 2 outlines the basics. Section 3 examines some solvability conditions of the quaternion equation presented in System (1.1) using the M−P inverses and rank equalities of the matrices, and derives the solution of System (1.1). Section 4 establishes some solvability conditions for matrix systems (1.2)–(1.4) to be solvable. Section 5 investigates some necessary and sufficient conditions for matrix equation (1.9) to have common solutions. Section 6 concludes the paper.
Marsaglia and Styan [18] presented the following rank equality lemma over the complex field, which can be generalized to H.
Lemma 2.1. [18] Let A∈Hm×n, B∈Hm×k, C∈Hl×n, D∈Hj×k, and E∈Hl×i be given. Then, the following rank equality holds:
r(ABLDREC0)=r(AB0C0E0D0)−r(D)−r(E). |
Lemma 2.2. [19] Let A∈Hm×n be given. Then,
(1)(Aη)†=(A†)η,(Aη∗)†=(A†)η∗;(2)r(A)=r(Aη∗)=r(Aη);(3)(LA)η∗=−η(LA)η=(LA)η=LAη∗=RAη∗,(4)(RA)η∗=−η(RA)η=(RA)η=RAη∗=LAη∗;(5)(AA†)η∗=(A†)η∗Aη∗=(A†A)η=Aη(A†)η;(6)(A†A)η∗=Aη∗(A†)η∗=(AA†)η=(A†)ηAη. |
Lemma 2.3. [20] Let A1 and A2 be given quaternion matrices with adequate shapes. Then, the equation A1X=A2 is solvable if, and only if, A2=A1A†1A2. In this case, the general solution to this equation can be expressed as
X=A†1A2+LA1U1, |
where U1 is any matrix with appropriate size.
Lemma 2.4. [20] Let A1 and A2 be given quaternion matrices with adequate shapes. Then, the equation XA1=A2 is solvable if, and only if, A2=A2A†1A1. In this case, the general solution to this equation can be expressed as
X=A2A†1+U1RA1, |
where U1 is any matrix with appropriate size.
Lemma 2.5. [21] Let A,B, and C be known quaternion matrices with appropriate sizes. Then, the matrix equation
AXB=C |
is consistent if, and only if,
RAC=0,CLB=0. |
In this case, the general solution to this equation can be expressed as
X=A†CB†+LAU+VRB, |
where U and V are any quaternion matrices with appropriate sizes.
Lemma 2.6. [15] Let Ci,Di, and Z(i=¯1,4) be known quaternion matrices with appropriate sizes.
C1X1+X2D1+C2Y1D2+C3Y2D3+C4Y3D4=Z. | (2.1) |
Denote
RC1C2=C12,RC1C3=C13,RC1C4=C14,D2LD1=D21,D31LD21=N32,D3LD1=D31,D4LD1=D41,RC12C13=M23,S12=C13LM23,RC1ZLD1=T1,C32=RM23RC12,A1=C32C14,A2=RC12C14,A3=RC13C14,A4=C14,D13=LD21LN32,B1=D41,B2=D41LD31,B3=D41LD21,B4=D41D13,E1=C32T1,E2=RC12T1LD31,E3=RC13T1LD21,E4=T1D13,A24=(LA2,LA4),B13=(RB1RB3),A11=LA1,B22=RB2,A33=LA3,B44=RB4,E11=RA24A11,E22=RA24A33,E33=B22LB13,E44=B44LB13,N=RE11E22,M=E44LE33,K=K2−K1,E=RA24KLB13,S=E22LN,K11=A2LA1,G1=E2−A2A†1E1B†1B2,K22=A4LA3,G2=E4−A4A†3E3B†3B4,K1=A†1E1B†1+LA1A†2E2B†2,K2=A†3E3B†3+LA3A†4E4B†4. |
Then, the following statements are equivalent:
(1) Equation (2.1) is consistent.
(2)
RAiEi=0,EiLBi=0(i=¯1,4),RE11ELE44=0. |
(3)
r(ZC2C3C4C1D10000)=r(D1)+r(C2,C3,C4,C1),r(ZC2C4C1D3000D1000)=r(C2,C4,C1)+r(D3D1),r(ZC3C4C1D2000D1000)=r(C3,C4,C1)+r(D2D1),r(ZC4C1D200D300D100)=r(D2D3D1)+r(C4,C1),r(ZC2C3C1D4000D1000)=r(C2,C3,C1)+r(D4D1),r(ZC2C1D300D400D100)=r(D3D4D1)+r(C2,C1),r(ZC3C1D200D400D100)=r(D2D4D1)+r(C3,C1),r(ZC1D20D30D40D10)=r(D2D3D4D1)+r(C1),r(ZC2C1000C4D3000000D1000000000−ZC3C1C4000D2000000D1000D400D4000)=r(D30D100D20D1D4D4)+r(C2C100C400C3C1C4). |
Under these conditions, the general solution to the matrix equation (2.1) is
X1=C†1(Z−C2Y1D2−C3Y2D3−C4Y3D4)−C†1U1D1+LC1U2,X2=RC1(Z−C2Y1D2−C3Y2D3−C4Y3D4)D†1+C1C†1U1+U3RD1,Y1=C†12TD†21−C†12C13M†23TD†21−C†12S12C†13TN†32D31D†21−C†12S12U4RN32D31D†21+LC12U5+U6RD21,Y2=M†23TD†31+S†12S12C†13TN†32+LM23LS12U7+U8RD31+LM23U4RN32,Y3=K1+LA2V1+V2RB1+LA1V3RB2, or Y3=K2−LA4W1−W2RB3−LA3W3RB4, |
where T=T1−C4Y3D4,Ui(i=¯1,8) are arbitrary matrices with appropriate sizes over H,
V1=(Im,0)[A†24(K−A11V3B22−A33W3B44)−A†24U11B13+LA24U12],W1=(0,Im)[A†24(K−A11V3B22−A33W3B44)−A†24U11B13+LA24U12],W2=[RA24(K−A11V3B22−A33W3B44)B†13+A24A†24U11+U21RB13](0In),V2=[RA24(K−A11V3B22−A33W3B44)B†13+A24A†24U11+U21RB13](In0),V3=E†11KE†33−E†11E22N†KE†33−E†11SE†22KM†E44E†33−E†11SU31RME44E†33+LE11U32+U33RE33,W3=N†KE44+S†SE†22KM†+LNLSU41+LNU31RM−U42RE44, |
U11,U12,U21,U31,U32,U33,U41, and U42 are arbitrary quaternion matrices with appropriate sizes, and m and n denote the column number of C4 and the row number of D4, respectively.
Some necessary and sufficient conditions for System (1.1) to be solvable will be established in this section. The general solution of System (1.1) will also be derived in this section. Moreover, we provide an example to illustrate our main results.
Theorem 3.1. Let Aii,Bii,Cii,Dii,Eii,Fii,H11, and G11 (i = 1, 2) be given quaternion matrices. Put
{A1=C11LA11,P1=E11−C11A†11B11D11,B2=RA22D22,P2=E22−C22B22A†22D22,^B1=RB2RA22F22,^A2=F11LA11LA1,^A3=F11LA11,^B3=RD11H11,^A4=LC22,^B4=RA22F22,H11L^B1=^B11,P=G11−F11A†11B11H11−F11LA11A†1P1D†11H11−B22A†22F22−C†22P2B†2RA22F22, | (3.1) |
{^B22L^B11=N1,^B3L^B1=^B22,^B4L^B1=^B33,R^A2^A3=^M1,S1=^A3L^M1,T1=PL^B1,C=R^M1R^A2,C1=C^A4,C2=R^A2^A4,C3=R^A3^A4,C4=^A4,D=L^B11LN1,D1=^B33,D2=^B33L^B22,D4=^B33D,E1=CT1,E2=R^A2T1L^B22,E3=R^A3T1L^B11,E4=T1D,^C11=(LC2,LC4),D3=^B33L^B11,^D11=(RD1RD3),^C22=LC1,^D22=RD2,^C33=LC3,^D33=RD4,^E11=R^C11^C22,^E22=R^C11^C22,^E33=^D22L^D11,^E44=^D33L^D11,M=R^E11^E22,N=^E44L^E33,F=F2−F1,E=R^C11FL^D11,S=^E22LM,^F11=C2LC1,G1=E2−C2C†1E1D†1D2,^F22=C4LC3,G2=E4−C4C†3E3D†3D4,F1=C†1E1D†1+LC1C†2E2D†2,F2=C†3E3D†3+LC3C†4E4D†4. | (3.2) |
Then, the following statements are equivalent:
(1) System (1.1) is solvable.
(2)
RA11B11=0,RA1P1=0,P1LD11=0,B22LA22=0,RC22P2=0,P2LB2=0,RCiEi=0,EiLDi=0(i=¯1,4),R^E11EL^E44=0. |
(3)
r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11), | (3.3) |
r(E11D11)=r(D11),r(B22A22)=r(A22), | (3.4) |
r(E22,C22)=r(C22),r(E22C22B22D22A22)=r(D22,A22), | (3.5) |
r(F220D22A22B11H11A1100C22G11C22F11E22C22B22)=r(F22,D22,A22)+r(A11C22F11), | (3.6) |
r(H110−D1100F2200D22A220C11E11000A11B11D1100C22G11C22F110E22C22B22)=r(C11A11C22F11)+r(H11D1100F220D22A22), | (3.7) |
r(H11000F220D22A220A1100C22G11C22F11E22C22B22)=r(H1100F22D22A22)+r(A11C22F11), | (3.8) |
r(H1100F22D22A22C22G11E22C22B22)=r(H1100F22D22A22), | (3.9) |
r(G11F11B22F220A22B11H11A110)=r(F11A11)+r(F22,A22), | (3.10) |
r(G11F110B22H110−D110F2200A220C11E1100A11B11D110)=r(H11D110F220A22)+r(F11C11A11), | (3.11) |
r(G11F11B22H1100F220A220A110)=r(H110F22A22)+r(F11A11), | (3.12) |
r(G11B22H110F22A22)=r(H110F22A22), | (3.13) |
r(H11000000D110F220000D22A220000H1100000000F22D22A220000F220F2200000A220C1100000−E1100A1100000−B11D110C22G11C22F11000E22C22B2200)=r(H1100000D110F22000D22A22000H110000000F22D22A220000F22F2200000A22)+r(C11A11C22F11). | (3.14) |
Proof. (1)⇔(2): The System (1.1) can be written as follows.
A11X1=B11, X2A22=B22, | (3.15) |
C11X1D11=E11, C22X2D22=E22, | (3.16) |
and
F11X1H11+X2F22=G11. | (3.17) |
Next, the solvability conditions and the expression for the general solutions of Eq (3.15) to Eq (3.17) are given by the following steps:
Step 1: According to Lemma 2.3 and Lemma 2.4, the system (3.15) is solvable if, and only if,
RA11B11=0, B22LA22=0. | (3.18) |
When condition (3.18) holds, the general solution of System (3.15) is
X1=A†11B11+LA11U1, X2=B22A†22+U2RA22. | (3.19) |
Step 2: Substituting (3.19) into (3.16) yields,
A1U1D11=P1, C22U2B2=P2, | (3.20) |
where A1,P1,B2,P2 are defined by (3.1). By Lemma 2.5, the system (3.20) is consistent if, and only if,
RA1P1=0, P1LD11=0, RC22P2=0, P2LB2=0. | (3.21) |
When (3.21) holds, the general solution to System (3.20) is
U1=A†1P1D†11+LA1W1+W2RD11,U2=C†22P2B†2+LC22W3+W4RB2. | (3.22) |
Comparing (3.22) and (3.19), hence,
X1=A†11B11+LA11A†1P1D†11+LA11LA1W1+LA11W2RD11,X2=B22A†22+C†22P2B†2RA22+LC22W3RA22+W4RB2RA22. | (3.23) |
Step 3: Substituting (3.23) into (3.17) yields
W4^B1+^A2W1H11+^A3W2^B3+^A4W3^B4=P, | (3.24) |
where ^Bi,^Aj(i=¯1,4,j=¯2,4) are defined by (3.1). It follows from Lemma 2.6 that Eq (3.24) is solvable if, and only if,
RCiEi=0,EiLDi=0(i=¯1,4),R^E11EL^E44=0. | (3.25) |
When (3.25) holds, the general solution to matrix equation (3.24) is
W1=^A2†T^B11†−^A2†^A3^M1†T^B11†−^A2†S1^A3†TN†1^B22^B11†−^A2†S1V4RN1^B22^B11†+L^A2V5+V6R^B11,W2=^M1†T^B22†+S†1S1^A3†TN†1+L^M1LS1V7+V8R^B22+L^M1V4RN1,W3=F1+LC2^V1+^V2RD1+LC1^V3RD2, or W3=F2−LC4V1−V2RD3−LC3V3RD4,W4=(P−^A2W1H11−^A3W2^B3−^A4W3^B4)^B1†+V3R^B1, |
where Ci,Ei,Di(i=¯1,4),^E11,^E44 are defined as (3.2), T=T1−^A4W3^B4,Vi(i=¯1,8) are arbitrary matrices with appropriate sizes over H,
^V1=(Im,0)[^C11†(F−^C22V3^D22−^C33^V3^D33)−^C11†U11^D11+L^C11U12],V1=(0,Im)[^C11†(F−^C22V3^D22−^C33^V3^D33)−^C11†U11^D11+L^C11U12],V2=[R^C11(F−^C22V3^D22−^C33^V3^D33)^D11†+^C11^C11†U11+U21R^D11](0In),^V2=[R^C11(F−^C22V3^D22−^C33^V3^D33)^D11†+^C11^C11†U11+U21R^D11](In0),^V3=^E11†F^E33†−^E11†^E22M†F^E33†−^E11†S^E22†FN†^E44^E33†−^E11†SU31RN^E44^E33†+L^E11U32+U33R^E33,V3=M†F^E44†+S†S^E22†FN†+LMLSU41+LMU31RN−U42R^E44, |
U11,U12,U21,U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of C22 and the row number of A22, respectively. We summarize that System (1.1) has a solution if, and only if, (3.18), (3.21), and (3.25) hold, i.e., the System (1.1) has a solution if, and only if, (2) holds.
(2)⇔(3): We prove the equivalence in two parts. In the first part, we want to show that (3.18) and (3.21) are equivalent to (3.3) to (3.5), respectively. In the second part, we want to show that (3.25) is equivalent to (3.6) to (3.14). It is easy to know that there exist X01,X02,U01, and U02 such that
A11X01=B11, X02A22=B22,A1U01D11=P1, C22U02B2=P2 |
holds, where
X01=A†11B11,U01=A†1P1D†11,X02=B22A†22,U02=C†22P2B†2, |
P1=E11−C11X01D11,P2=E22−C22X02D22, and P=G11−F11X01H11−F11LA11U01H11−X02F22−U02RA22F22.
Part 1: We want to show that (3.18) and (3.21) are equivalent to (3.3) to (3.5), respectively. It follows from Lemma 2.1 and elementary transformations that
(3.18)⇔r(RA11B11)=0⇔r(B11,A11)=r(A11)⇔(3.3),(3.21)⇔r(RA1P1)=0⇔r(P1,A1)=r(A1)⇔r(E11−C11A†11B11D11,C11LA11)=r(C11LA11)⇔r(E11C11B11D11A11)=r(C11A11)⇔(3.3),(3.21)⇔r(P1LD11)=0⇔r(P1D11)=r(D11)⇔r(E11−C11A†11B11D11D11)=r(D11)⇔r(E11D11)=r(D11)⇔(3.4),(3.18)⇔r(B22LA22)=0⇔r(B22A22)=r(A22)⇔(3.4). |
Similarly, we can show that (3.21) is equivalent to (3.5). Hence, (3.18) and (3.21) are equivalent to (3.3) and (3.5), respectively.
Part 2: In this part, we want to show that (3.25) is equivalent to (3.6) and (3.14). According to Lemma 2.6, we have that (3.25) is equivalent to the following:
r(P^A2^A3^A4^B1000)=r(^B1)+r(^A2,^A3,^A4), | (3.26) |
r(P^A2^A4^B300^B100)=r(^A2,^A4)+r(^B3^B1), | (3.27) |
r(P^A3^A4H1100^B100)=r(^A3,^A4)+r(H11^B1), | (3.28) |
r(P^A4H110^B30^B10)=r(H11^B3^B1)+r(^A4), | (3.29) |
r(P^A2^A3^B400^B100)=r(^A2,^A3)+r(^B4^B1), | (3.30) |
r(P^A2^B30^B40^B10)=r(^B3^B4^B1)+r(^A2), | (3.31) |
r(P^A3H110^B40^B10)=r(H11^B4^B1)+r(^A3), | (3.32) |
r(PH11^B3^B4^B1)=r(H11^B3^B4^B1), | (3.33) |
r(P^A200^A4^B30000^B1000000−P^A3^A400H110000^B100^B40^B400)=r(^B30^B100H110^B1^B4^B4)+r(^A20^A40^A3^A4), | (3.34) |
respectively. Hence, we only prove that (3.26)–(3.34) are equivalent to (3.6)–(3.14) when we prove that (3.25) is equivalent to (3.6)–(3.14). Now, we prove that (3.26)–(3.34) are equivalent to (3.6)–(3.14). In fact, we only prove that (3.26), (3.30), and (3.34) are equivalent to (3.6), (3.10), and (3.14); the remaining part can be proved similarly. According to Lemma 2.1 and elementary transformations, we have that
(3.26)=r(P^A2^A3^A4^B1000)=r(^B1)+r(^A2,^A3,^A4)⇔r(G11−F11X01H11−F11LA11U01H11−X02F22−U02RA22F22F11LA11LA1F11LA11LC22RB2RA22F22000)=r(RB2RA22F22)+r(F11LA11LA1,F11LA11,LC22)⇔r(G11−F11X01H11−X02F22−U02RA22F22F11I0RA22F2200B20A110000C220)=r(RA22F22,B2)+r(F11IA1100C22)⇔r(G11F11IU02B20F2200B2A22B11H11A11000C22X02F220C2200)=r(F22,D22,A22)+r(F11IA1100C22)⇔r(F220D22A22B11H11A1100C22G11C22F11E22C22B22)=r(F22,D22,A22)+r(A11F11C22)⇔(3.6). |
Similarly, we have that (3.27)⇔(3.7),(3.28)⇔(3.8),(3.29)⇔(3.9).
(3.30)=r(P^A2^A3^B400^B100)=r(^A2,^A3)+r(^B4^B1)⇔r(G11−F11X01H11−F11LA11U01H11−X02F22−U02RA22F22F11LA11LA1F11LA1RA22F2200RB2RA22F2200)=r(F11LA11LA1,F11LA11)+r(RA22F22RB2RA22F22)⇔r(G11−F11X01H11F11B22F220A220A110)=r(F11A11)+r(F22,A22)⇔r(G11F11B22F220A22B11H11A110)=r(F11A11)+r(F22,A22)⇔(3.10). |
Similarly, we have that (3.31)⇔(3.11),(3.32)⇔(3.12),(3.33)⇔(3.13).
(3.34)=r(P^A200^A4^B30000^B1000000−P^A3^A400H110000^B100^B40^B400)=r(^B30^B100H110^B1^B4^B4)+r(^A20^A40^A3^A4)⇔r(PF11LA11LA100LC22RD11H110000RB2RA22F22000000−PF11LA11LC2200H110000RB2RA22F2200RA22F220RA22F2200)=r(RD11H110RB2RA22F2200H110RB2RA22F22RA22F22RA22F22)+r(F11LA11LA10LC220F11LA11LC22)⇔r(PF11LA1100LC22000H110000D1100RA22F2200000B2000−G11+X02F22+U02RA22F22F11LA11LC2200000H110000000RA22F220000B2RA22F220RA22F22000000A1000000)=r(H110D1100RA2200B200H110000RA22F2200B2RA22F22RA22F22000)+r(F11LA110LC220F11LA11LC22A100)⇔r(H11000000D110F220000D22A220000H1100000000F22D22A220000F220F2200000A220C1100000−E1100A1100000−B11D110C22G11C22F11000E22C22B2200)=r(H1100000D110F22000D22A22000H110000000F22D22A220000F22F2200000A22)+r(C11A11C22F11)⇔(3.14). |
Theorem 3.2. Let System (1.1) be solvable. Then, the general solution of System (1.1) is
X1=A†11B11+LA11A†1P1D†11+LA11LA1W1+LA11W2RD11,X2=B22A†22+C†22P2B†2RA22+LC22W3RA22+W4RB2RA22, |
where
W1=^A2†T^B11†−^A2†^A3^M1†T^B11†−^A2†S1^A3†TN†1^B22^B11†−^A2†S1V4RN1^B22^B11†+L^A2V5+V6R^B11,W2=^M1†T^B22†+S†1S1^A3†TN†1+L^M1LS1V7+V8R^B22+L^M1V4RN1,W3=F1+LC2^V1+^V2RD1+LC1^V3RD2, or W3=F2−LC4V1−V2RD3−LC3V3RD4,W4=(P−^A2W1H11−^A3W2^B3−^A4W3^B4)^B1†+V3R^B1,^V1=(Im,0)[^C11†(F−^C22V3^D22−^C33^V3^D33)−^C11†U11^D11+L^C11U12],V1=(0,Im)[^C11†(F−^C22V3^D22−^C33^V3^D33)−^C11†U11^D11+L^C11U12],V2=[R^C11(F−^C22V3^D22−^C33^V3^D33)^D11†+^C11^C11†U11+U21R^D11](0In),^V2=[R^C11(F−^C22V3^D22−^C33^V3^D33)^D11†+^C11^C11†U11+U21R^D11](In0),^V3=^E11†F^E33†−^E11†^E22M†F^E33†−^E11†S^E22†FN†^E44^E33†−^E11†SU31RN^E44^E33†+L^E11U32+U33R^E33,V3=M†F^E44†+S†S^E22†FN†+LMLSU41+LMU31RN−U42R^E44, |
T=T1−^A4W3^B4,Vi(i=¯4,8) are arbitrary matrices with appropriate sizes over H, U11,U12,U21, U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of C22 and the row number of A22, respectively.
Next, we consider a special case of the System (1.1).
Corollary 3.3. [16] Let Aii,Bii,Cii,Dii,Eii,Fii(i=1,2), and G11 be given matrices with appropriate dimensions over H. Denote
T=C11LA11,K=RA22D22, B1=RKRA22F22,A1=F11LA11LT,C3=F11LA11,D3=RD11,C4=LC22,D4=RA22F22,Aα=RA1C3,Bβ=D3LB1,Cc=RAαC4,Dd=D4LB1,E=RA1E1LB1,A=A†11B11+LA11T†(E11−C11A†11B11D11)D†,B=B22A†22+C†22(E22−C22B22A†22D22)K†RA22,E1=G11−F11A−BF22,M=RAαCc,N=DdLBβ,S=CcLM. |
Then, the following statements are equivalent:
(1) Equation (1.8) is consistent.
(2)
RA11B11=0,B22LA22=0,RC22E22=0,E11LD11=0,RT(E11−C11A†11B11D11)=0,(E22−C22B22A†22D22)LK=0,RMRAαE=0,ELBβLN=0,RAαELDd=0,RCcELBβ=0. |
(3)
r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11),r(B22A22)=r(A22),r(E22,C22)=r(C22),r(E22C22B22D22A22)=r(D22,A22),r(F220D22A22B11A1100C22G11C22F11E22C22B22)=r(F22,D22,A22)+r(A11C22F11),r(0F22D11D22A22C11E1100A11B11D1100C22F11C22G11D11E22C22B22)=r(C11A11C22F11)+r(F22D11,D22,A22),r(G11F11B22F220A22B11A110)=r(F11A11)+r(F22,A22),r(F11G11D11B220F22D11A22C11E110A11B11D110)=r(F22D11,A22)+r(F11C11A11). |
Finally, we provide an example to illustrate the main results of this paper.
Example 3.4. Conside the matrix equation (1.1)
A11=(a111a121),B11=(b111b112b121b122),C11=(c111c121),D11=(d111d121),E11=(e111e121),A22=(a211a212),B22=(b211b212b221b222),C22=(c211c212c221c222),D22=(d211),E22=(e211e221),F11=(f111f121),H11=(h111h112h121h122),F22=(f211f212),G11=(g111g112g121g122), |
where
a111=0.9787+0.5005i+0.0596j+0.0424k,a121=0.7127+0.4711i+0.6820j+0.0714k,b111=0.5216+0.8181i+0.7224j+0.6596k,b112=0.9730+0.8003i+0.4324j+0.0835k,b121=0.0967+0.8175i+0.1499j+0.5186k,b122=0.6490+0.4538i0.8253j+0.1332k,c111=0.1734+0.8314i+0.0605j+0.5269k,c121=0.3909+0.8034i+0.3993j+0.4168k,d111=0.6569+0.2920i+0.0159j+0.1671k,d121=0.6280+0.4317i+0.9841j+0.1062k,e111=0.3724+0.4897i+0.9516j+0.0527k,e121=0.1981+0.3395i+0.9203j+0.7379k,a211=0.2691+0.4228i+0.5479j+0.9427k,a212=0.4177+0.9831i+0.3015j+0.7011k,b211=0.6663+0.6981i+0.1781j+0.9991k,b212=0.0326+0.8819i+0.1904j+0.4607k,b221=0.5391+0.6665i+0.1280j+0.1711k,b222=0.5612+0.6692i+0.3689j+0.9816k,c211=0.1564+0.6448i+0.1909j+0.4820k,c212=0.5895+0.3846i+0.2518j+0.6171k,c221=0.8555+0.3763i+0.4283j+0.1206k,c222=0.2262+0.5830i+0.2904j+0.2653k,d211=0.8244+0.9827i+0.7302j+0.3439k,e211=0.5847+0.9063i+0.8178j+0.5944k,e221=0.1078+0.8797i+0.2607j+0.0225k,f111=0.4253+0.1615i+0.4229j+0.5985k,f121=0.3127+0.1788i+0.0942j+0.4709k,h111=0.6959+0.6385i+0.0688j+0.5309k,h112=0.4076+0.7184i+0.5313j+0.1056k,h121=0.6999+0.0336i+0.3196j+0.6544k,h122=0.8200+0.9686i+0.3251j+0.6110k,f211=0.7788+0.4235i+0.0908j+0.2665k,f212=0.1537+0.2810i+0.4401j+0.5271k,g111=0.4574+0.5181i+0.6377j+0.2407k,g112=0.2891+0.6951i+0.2548j+0.6678k,g121=0.8754+0.9436i+0.9577j+0.6761k,g122=0.6718+0.0680i+0.2240j+0.8444k. |
Computing directly yields the following:
r(B11A11)=r(A11)=2,r(E11C11B11D11A11)=r(C11A11)=2,r(E11D11)=r(D11)=1,r(B22A22)=r(A22)=2,r(E22C22)=r(C22)=2,r(E22C22B22D22A22)=r(D22A22)=3,r(F220D22A22B11H11A1100C22G11C22F11E22C22B22)=r(F22D22A22)+r(A11C22F11)=5,r(H110−D1100F2200D22A220C11E11000A11B11D1100C22G11C22F110E22C22B22)=r(C11A11C22F11)+r(H11D1100F220D22A22)=7,r(H11000F220D22A220A1100C22G11C22F11E22C22B22)=r(H1100F22D22A22)+r(A11C22F11)=6,r(H1100F22D22A22C22G11E22C22B22)=r(H1100F22D22A22)=5,r(G11F11B22F220A22B11H11A110)=r(F11A11)+r(F22,A22)=5,r(G11F110B22H110−D110F2200A220C11E1100A11B11D110)=r(H11D110F220A22)+r(F11C11A11)=6,r(G11F11B22H1100F220A220A110)=r(H110F22A22)+r(F11A11)=5, r(G11B22H110F22A22)=r(H110F22A22)=4,r(H11000000D110F220000D22A220000H1100000000F22D22A220000F220F2200000A220C1100000−E1100A1100000−B11D110C22G11C22F11000E22C22B2200)=r(H1100000D110F22000D22A22000H110000000F22D22A220000F22F2200000A22)+r(C11A11C22F11)=11. |
All rank equations in (3.3) to (3.14) hold. So, according to Theorem 3.1, the system of matrix equation (1.1) has a solution. By Theorem 3.2, the solution of System (1.1) can be expressed as
X1=(0.4946+0.1700i−0.1182j−0.3692k0.4051−0.0631i−0.2403j−0.1875k),X2=(−0.0122+0.2540i−0.3398j−0.3918k0.7002−0.3481i−0.2169j+0.0079k). |
In this section, we use the same method and technique as in Theorem 3.1 to study the three systems of Eqs (1.2)–(1.4). We only present their results and omit their proof.
Theorem 4.1. Consider the matrix equation (1.2) over H, where Aii,Bii,Cii,Dii,Eii,Fii,G11, and H11(i=¯1,2) are given. Put
A1=C11LA11,P1=E11−C11A†11B11D11,B2=RA22D22,P2=E22−C22B22A†22D22,^A1=F11LA11LA1,^A2=F11LA1,^B2=RD11,^A3=H11LC22,^B3=RA22F22,^B4=RB2RA22F22,B=G11−F11A†11B11−F11LA11A†1P1D†11−H11B22A†22F22−H11C†22P2B†2RA22F22,R^A1^A2=A12,R^A1^A3=A13,R^A1H11=A14,^B3L^B2=N1,RA12A13=M1,S1=A13LM1,R^A1B=T1,C=RM1RA12,^C1=CA14,^C2=RA12A14,^C3=RA13A14,^C4=A14,D=L^B2LN1,^D1=^B4,^D2=^B4L^B3,^D3=^B4L^B2,^D4=^B4D,^E1=CT1,^E2=RA12T1L^B3,^E3=RA13T1L^B2,^E4=T1D,C24=(L^C2,L^C4),D13=(R^D1R^D3),C12=L^C1,D12=R^D2,C33=L^C3,D33=R^D4,E24=RC24C12,E13=RC24C33,E33=D12LD13,E44=D33LD13,M=RE24E13,N=E44LE33,F=F2−F1,E=RC24FLD13,S=E13LM,^F11=^C2L^C1,^G1=^E2−^C2^C1†^E1^D1†^D2,F33=^C4L^C3,^G2=^E4−^C4^C3†^E3^D3†^D4,F1=^C1†^E1^D1†+L^C1^C2†^E2^D2†,F2=^C3†^E3^D3†+L^C3^C4†^E4^D4†. |
Then, the following statements are equivalent:
(1) System (1.2) is consistent.
(2)
RA11B11=0,RA1P1=0,P1LD11=0,B22LA22=0,RC22P2=0,P2LB2=0,R^Ci^Ei=0,^EiL^Di=0(i=¯1,4),RE24ELE44=0. |
(3)
r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11),r(B22,A22)=r(A22),r(E22,C22)=r(C22),r(E22C22B22D22A22)=r(D22,A22),r(G11D11F11H11E11C110B11D11A110)=r(F11H11C110A110),r(G11D11F11H110F22D1100A22E11C1100B11D11A1100)=r(F22,A22)+r(F11H11C110A110),r(H11F11G11D110C11E110A11B11D11)=r(H11F110C110A11),r(H11F110G11D1100A22F22D110C110E110A110B11D11)=r(F22D11,A22)+r(H11F110C110A11),r(G11D11F11H1100F22D1100D22A22E11C1100000C22−E22−C22B22B11D11A11000)=r(F11H11C1100C22A110)+r(F22,D22,A22),r(G11D11F11H11B22F22D110A22E11C110B11D11A110)=r(F11C11A11)+r(F22,A22),r(H11F1100G11D1100D22A22F22D110C1100E110A1100B11D11C220−E22−C22B220)=r(H11F110C110A11C220)+r(D22,A22,F22D11),r(F11H11B22G11D110A22F22D11C110E11A110B11D11)=r(F11C11A11)+r(A22,F22D11),r(G11F1100H1100H5B220F22000000A22000H11F11H110−H11B220G11D1100000D22A220−F22D1100C2200E22000000C110000E11000A110000B11D11B11A110000000)=r(F2200A2200D22A220F22D11)+r(F1100H110H11F11H110C220000C11000A110A11000). |
Under these conditions, the general solution of System (1.2) is
X1=A†11B11+LA11A†1P1D†11+LA11LA1W1+LA11W2RD11,X2=B22A†22+C†22P2B†2RA22+LC22W3RA22+W4RB2RA22, |
where
W1=^A1†(B−^A2W1^B2−^A3W3^B3−H11W4^B4)+L^A1U1,W2=A†12T^B2†−A†12A13M†1T^B2†−A†12S1A†13TN†1^B3^B2†−A†12S1U2RN1^B3^B2†+LA12U3+U4R^B2,W3=M†1T^B3†+S†1S1A†13TN†1+LM1LS1U5+U6R^B3+LM1U2RN1,W4=F1+L^C2V1+V2R^D1+L^C1V3R^D2, or W4=F2−L^C4^V1−^V2R^D3−L^C3^V3R^D4, |
where T=T1−H11W4^B4,Ui(i=¯1,6) are arbitrary matrices with appropriate sizes over H,
V1=(Im,0)[C†24(F−C12V3D12−C33^V3D33)−C†24U11D13+LC24U12],^V1=(0,Im)[C†24(F−C12V3D12−C33^V3D33)−C†24U11D13+LC24U12],^V2=[RC24(F−C12V3D12−C33^V3D33)D†13+C24C†24U11+U21RD13](0In),V2=[RC24(F−C12V3D12−C33^V3D33)D†13+C24C†24U11+U21RD13](In0),V3=E†24FE†33−E†24E13M†FE†33−E†24SE†13FN†E44E†33−E†24SU31RNE44E†33+LE24U32+U33RE33,^V3=M†FE†44+S†SE†13FN†+LMLSU41+LMU31RN−U42RE44, |
U11,U12,U21,U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of H11 and the row number of A22, respectively.
Theorem 4.2. Consider the matrix equation (1.3) over H, where Aii,Bii,Cii,Dii,Eii,Fii,G11H11(i=¯1,2) are given. Put
A1=C11LA11,P1=E11−C11A†11B11D11,A2=C22LA22,P2=E22−C22A†22B22D22,^A1=F11LA11LA1,^A2=F11LA11,^B2=RD11,^A11=H11LA22LA2,^A22=H11LA22,^B4=RD22F22,B=G11−F11A†11B11−F11LA11A†1P1D†11−H11A†22B22F22−H11LA22A†2P2D†22F22,R^A1^A2=A12,R^A1^A11=A13,R^A1^A22=A33,F22L^B2=N1,RA12A13=M1,S1=A13LM1,R^A1B=T1,C=RM1RA12,^C1=CA33,^C2=RA12A33,^C11=RA13A33,^C22=A33,D=L^B2LN1,^D1=^B4,^D2=^B4LF22,^D11=^B4L^B2,^D22=^B4D,^E1=CT1,^E2=RA12T1LF22,^E11=RA13T1L^B2,^E4=T1D,C24=(L^C2,L^C22),D13=(R^D1R^D11),C21=L^C1,D12=R^D2,C33=L^C11,D33=R^D22,E11=RC24C21,E22=RC24C33,E33=D12LD13,E44=D33LD13,M=RE11E22,N=E44LE33,F=F2−F1,E=RC24FLD13,S=E22LM,^F11=^C2L^C1,^G1=^E2−^C2^C1†^E1^D1†^D2,^F22=^C22L^C11,^G2=^E4−^C22^C11†^E11^D11†^D22,F1=^C1†^E1^D1†+L^C1^C2†^E2^D2†,F2=^C11†^E11^D11†+L^C11^C22†^E4^D22†. |
Then, the following statements are equivalent:
(1) System (1.3) is consistent.
(2)
RA11B11=0,RA1P1=0,P1LD11=0,RA22B22=0,RA2P2=0,P2LD22=0,R^Ci^Ei=0,R^C11^E11=0,R^C22^E4=0,^EiL^Di=0(i=¯1,2),^E11L^D11=0,^E4L^D22=0,RE11ELE44=0. |
(3)
r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11),r(B22,A22)=r(A22),r(E22C22B4D22A22)=r(C22A22),r(E22D22)=r(D22),r(G11F11H11B11A110B22F220A22)=r(F11H11A1100A22),r(G11F11H11F2200B11A11000A22)=r(F22)+r(F11H11A1100A22),r(H11F11G11D11A220B22F22D110C11E110A11B11D11)=r(H11F110C110A11A220),r(H11F11G11D1100F22D110C11E110A11B11D11A2200)=r(H11F110C110A11A220)+r(F22D11),r(G11F11H110F2200D22B11A110000C22−E2200A22−B22D22)=r(F11H11A1100C220A22)+r(F22,D22),r(G11F11F220B11A11)=r(F11A11)+r(F22),r(H11F110G11D1100D22F22D11C220−E2200C110E11A2200B22F22D110A110B11D11)=r(H11F11C2200C11A2200A11)+r(D22,F22D11),r(F11G11D110F22D11C11E11A11B11D11)=r(F11C11A11)+r(F22D11),r(G11F11000H110F2200000000−G11D11H11F11H11000F22D11000B22B11A1100000000C2200E2200−E110C110000−B22F22D11A2200000−B11D110A110000000A220)=r(F22000D22F22D11)+r(F1100H110H11F11H110C22000A220000C11000A110A11000000A22). |
Under these conditions, the general solution of System (1.3) is
X1=A†11B11+LA11A†1P1D†11+LA11LA1W1+LA11W2RD11,X2=A†22B4+LA22A†2P2D†22+LA22LA2W3+LA22W4RD22, |
where
W1=^A1†(B−^A2W1^B2−^A11W3F22−^A22W4^B4)+L^A1U1,W2=A†12T^B2†−A†12A13M†1T^B2†−A†12S1A†13TN†1F22^B2†−A†12S1U2RN1F22^B2†+LA12U3+U4R^B2,W3=M†1TF†22+S†1S1A†13TN†1+LM1LS1U5+U6RF22+LM1U2RN1,W4=F1+L^C2V1+V2R^D1+L^C1V3R^D2, or W4=F2−L^C22^V1−^V2R^D11−L^C11^V3R^D22, |
where T=T1−^A22W4^B4,Ui(i=¯1,6) are arbitrary matrices with appropriate sizes over H,
V1=(Im,0)[C†24(F−C21V3D12−C33^V3D33)−C†24U11D13+LC24U12],^V1=(0,Im)[C†24(F−C21V3D12−C33^V3D33)−C†24U11D13+LC24U12],^V2=[RC24(F−C21V3D12−C33^V3D33)D†13+C24C†24U11+U21RD13](0In),V2=[RC24(F−C21V3D12−C33^V3D33)D†13+C24C†24U11+U21RD13](In0),V3=E†11FE†33−E†11E22M†FE†33−E†11SE†22FN†E44E†33−E†11SU31RNE44E†33+LE11U32+U33RE33,^V3=M†FE†44+S†SE†22FN†+LMLSU41+LMU31RN−U42RE44, |
U11,U12,U21,U31,U32,U33,U41, and U42 are any matrices with appropriate sizes, and m and n denote the column number of H11 and the row number of D22, respectively.
Theorem 4.3. Consider the matrix equation (1.4) over H, where Aii,Bii,Cii,Dii,Eii,Fii(i=¯1,2), and G11 are given. Put
^A1=C11LA11,P1=E11−C11A†11B11D11,^A2=C22LA22,P2=E22−C22A†22B22D22,A5=F11LA1L^A1,A6=F11LA11,A7=LA22L^A2,A8=LA22,B5=RD11,B7=RD22F22,B=G11−F11A†11B11−F11LA1^A1†P1D†11−A†22B22F22−LA22^A2†P2D†22F22,RA5A6=A11,RA5A7=A2,RA5A8=A33,F22LB5=N1,RA11A2=M1,S1=A2LM1,RA5B=T1,C=RM1RA11,^C1=CA33,^C2=RA11A33,^C11=RA2A33,^C4=A33,D=LB5LN1,^D1=B7,^D2=B7LF22,^D3=B7LB5,^D4=B7D,^E1=CT1,^E2=RA11T1LF22,^E11=RA2T1LB5,^E4=T1D,C1=(L^C2,L^C4),D13=(R^D1R^D3),D1=L^C1,D2=R^D2,C33=L^C11,D33=R^D4,E11=RC1D1,E2=RC1C33,E33=D2LD13,E44=D33LD13,M=RE11E2,N=E44LE33,F=^F2−^F1,E=RC1FLD13,S=E2LM,F11=^C2L^C1,^G1=^E2−^C2^C1†^E1^D1†^D2,F33=^C4L^C11,^G2=^E4−^C4^C11†^E11^D3†^D4,^F1=^C1†^E1^D1†+L^C1^C2†^E2^D2†,^F2=^C11†^E11^D3†+L^C11^C4†^E4^D4†. |
Then, the following statements are equivalent:
(1) Equation (1.4) is consistent.
(2)
RA11B11=0,R^A1P1=0,P1LD11=0,RA22B22=0,R^A2P2=0,P2LD22=0, R^Ci^Ei=0,^EiL^Di=0(i=¯1,2),R^C11^E11=0,R^C4^E4=0,^E11L^D3=0,^E4L^D4=0,RE11ELE44=0. |
(3)
r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11), r(B22,A22)=r(A22),r(E22C22B22D22A22)=r(C22A22),r(E22D22)=r(D22),r(B11A11A22G11−B22F22A22F11)=r(A11A22F11),r(F220B11A11A22G11A22F11)=r(F22)+r(A11A22F11),r(C11E11A11B11D11−A22F11B22F22D11−A22G11D11)=r(C11A11A22F11),r(0F22D11C11E11A11B11D11A22F11A22G11D11)=r(C11A11A22F11)+r(F22D11),r(F220D22C22G11C22F11E22B11A110A22G11A22F11B22D22)=r(F22,D22)+r(C22F11A22F11A11),r(G11F11F220B11A11)=r(F11A11)+r(F22),r(0D22F22D11C22F11E22C22G11D11C110E22A22F110A22G11D11−B22F22D11A110B11D11)=r(C22F11C11A22F11A11)+r(D22,F22D11),r(F11G11D110F22D11C11E11A11B11D11)=r(F11C11A11)+r(F22D11), |
r(F22000000F22D110B22B11A11000C22G11C22F11C22G11D11−C22F11E2200−E11C110A22G11A22F11A22G11D11−B22F22D11−A22F11000−B11D11A110A22G11A22F11000)=r(F22000F22D11D22)+r(−C22F11C22F11−A22F11A22F110C110A11A110A110−A22F110). |
Under these conditions, the general solution of System (1.4) is
X1=A†11B11+LA1^A1†P1^B1†+LA1L^A1W1+LA1W2R^B1,X2=A†2B22+LA2^A2†P2^B2†+LA2L^A2W3+LA3W4R^B2, |
where
W1=A†5(B−A6W1B5−A7W3F22−A8W4B7)+LA5U1,W2=A†1TB†5−A†1A2M†1TB†5−A†1S1A†2TN†1F22B†5−A†1S1U2RN1F22B†5+LA1U3+U4RB5,W3=M†1TF†22+S†1S1A†2TN†1+LM1LS1U5+U6RF22+LM1U2RN1,W4=^F1+L^C2V1+V2R^D1+L^C1V3R^D2, or W4=^F2−L^C4^V1−^V2R^D3−L^C11^V3R^D4, |
where T=T1−A8W4B7,Ui(i=¯1,6) are arbitrary matrices with appropriate sizes over H,
V1=(Im,0)[C†1(F−D1V3D2−C33^V3D33)−C†1U11D1+LC1U12],^V1=(0,Im)[C†1(F−D1V3D2−C33^V3D33)−C†1U11D1+LC1U12],^V2=[RC1(F−D1V3D2−C33^V3D33)D†1+C1C†1U11+U21RD1](0In),V2=[RC1(F−C2V3D2−C33^V3D33)D†1+C1C†1U11+U21RD1](In0),V3=E†11FE†33−E†11E2M†FE†33−E†11SE†2FN†E44E†33−E†11SU31RNE44E†33+LE11U32+U33RE33,^V3=M†FE†44+S†SE†2FN†+LMLSU41+LMU31RN−U42RE44, |
U11,U12,U21,U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of A22 and the row number of D22, respectively.
In this section, we use the Lemma 2.2 and the Theorem 3.1 to study matrix equation (1.9) involving η-Hermicity matrices.
Theorem 5.1. Let A11,B11,C11,E11,F11,F22, and G11(G11=Gη∗11) be given matrices. Put
A1=C11LA11,P1=E11−C11A†11B11Cη∗11,B2=Aη∗1,P2=Pη∗1,ˆB1=RB2(F22LA11)η∗,ˆA3=F11LA11,ˆA2=ˆA3LA1,ˆA4=LC11,ˆB3=(F11ˆA4)η∗,ˆB4=(F22LA11)η∗,Fη∗11LˆB1=ˆB11,P=G11−F11A†11B11Fη∗11−ˆA3A†1P1(F11C†11)η∗−(F22A†11B11)η∗−C†11P2B†2ˆB4,ˆB22LB11=N1,ˆB3LˆB1=ˆB22,ˆB4LˆB1=ˆB33,RˆA2ˆA3=ˆM1,S1=ˆA3LM1,T1=PL^B1,C=RM1RˆA2,C1=CˆA4,C2=RˆA2ˆA4,C3=RˆA3ˆA4,C4=ˆA4,D=LˆB11LN1,D1=ˆB33,D2=ˆB33LˆB22,D4=ˆB33D,E1=CT1,E2=RˆA2T1LˆB11,E4=T1D,ˆC11=(LC2,LC4),D3=ˆB33LˆB11,ˆD11=(RD1RD3),ˆC22=LC1,ˆD22=RD2, ˆC33=LC3,ˆD33=RD4,ˆE11=RˆC11ˆC22,ˆE22=RˆC11ˆC33,ˆE33=ˆD22LˆD11,ˆE44=ˆD33LˆD11,M=RˆE11ˆE22,N=ˆE44LˆE33, F=F2−F1,E=RˆC11FLˆD11,S=ˆE22LM,^F11=C2LC1,G1=E2−C2C†1E1D†1D2,^F22=C4LC3,G2=E4−C4C†3E3D†3D4,F1=C†1E1D†1+L†C1C†2E2D†2,F2=C†3E3D†3+LC3C†4E4D†4. |
Then, the following statements are equivalent:
(1) System (1.9) is solvable.
(2)
RA11B11=0,RA1P1=0,P1(RC11)η∗=0,RCiEi=0,EiLDi=0(i=¯1,4),RˆE11ELˆE44=0. |
(3)
r(B11,A11)=r(A11),r(E11C11B11Cη∗11A11)=r(C11A11), r(E11Cη∗11)=r(C11),r(Fη∗220Cη∗11Aηη∗11B11Fη∗11A1100C11G11C11F11Eη∗11C11Bη∗11)=r(Fη∗22,Cη∗11,Aη∗11)+r(A11C11F11),r(Fη∗110−Cη∗1100Fη∗2200Cη∗11Aη∗110C11E11000A11B11Cη∗1100C11G11C11F110Eη∗11C11Bη∗11)=r(C11A110)+r(Fη∗11Cη∗1100Fη∗220Cη∗11Aη∗11),r(Fη∗11000Fη∗220Cη∗11Aη∗110A1100C11G11C11F11Eη∗11C11Bη∗11)=r(Fη∗1100Fη∗22Cη∗11Aη∗11)+r(A11C11F11),r(Fη∗1100Fη∗22Cη∗11Aη∗11C11G11Eη∗11C11Bη∗11)=r(Fη∗1100Fη∗22Cη∗11Aη∗11,),r(G11F11Bη∗11Fη∗220Aη∗11B11Fη∗11A110)=r(F11A11)+r(Fη∗22,Aη∗11),r(G11F110Bη∗11Fη∗110−Cη∗110Fη∗2200Aη∗110C11E1100A11B11Cη∗110)=r(Fη∗11Cη∗110Fη∗220Aη∗11)+r(F11C11A11),r(G11F11Bη∗11Fη∗1100Fη∗220Aη∗110A110)=r(Fη∗110Fη∗22Aη∗11)+r(F11A11),r(G11Bη∗11Fη∗110Fη∗22Aη∗11)=r(Fη∗110Fη∗22Aη∗11),r(Fη∗11000000Cη∗110Fη∗220000Cη∗11Aη∗110000Fη∗1100000000Fη∗22Cη∗11Aη∗110000Fη∗220Fη∗2200000Aη∗110C1100000−E1100A1100000−B11Cη∗110C11G11C11F11000Eη∗11C11Bη∗1100)=r(Fη∗1100000Cη∗110Fη∗22000Cη∗11Aη∗11000Fη∗110000000Fη∗22Cη∗11Aη∗110000Fη∗22Fη∗2200000Aη∗11)+r(C11A11C11F11). |
Proof. Evidently, the system of Eq (1.9) has a solution if and only if the following matrix equation has a solution:
A11^X1=B11,C11^X1Cη∗11=E11,^X2Aη∗11=Bη∗11,C11^X2Cη∗11=Eη∗11,F11X1Fη∗11+^X2η∗Fη∗22=G11. | (5.1) |
If (1.9) has a solution, say, X1, then (^X1, ^X2):=(X1, Xη∗1) is a solution of (5.1). Conversely, if (5.1) has a solution, say (^X1, ^X2), then it is easy to show that (1.5) has a solution
X1:=^X1+Xη∗22. |
According to Theorem 3.1, we can deduce that this theorem holds.
We have established the solvability conditions and the expression of the general solutions to some constrained systems (1.1)–(1.4). As an application, we have investigated some necessary and sufficient conditions for Eq (1.9) to be consistent. It should be noted that the results of this paper are valid for the real number field and the complex number field as special number fields.
Long-Sheng Liu, Shuo Zhang and Hai-Xia Chang: Conceptualization, formal analysis, investigation, methodology, software, validation, writing an original draft, writing a review, and editing. All authors of this article have contributed equally. All authors have read and approved the final version of the manuscript for publication.
This work is supported by the National Natural Science Foundation(No. 11601328) and Key scientific research projects of univesities in Anhui province(No. 2023AH050476).
The authors declare that they have no conflicts of interest.
[1] | Lacasse S, Berre T, Lefebvre G (1985) Block sampling of sensitive clay. In: Publications Committee of XI ICSMFE, editor. Proceedings of the Eleventh International Conference on Soil Mechanics and Foundation Engineering, San Francisco, 12-16 August 1985, Rotterdam: A.A. Balkema, 887–892. |
[2] | Lunne T, Long M, Forsberg CF (2003) Characterization and engineering properties of Onsøy clay. In: Tan TS, Phoon KK, Hight DW, et al., editors. Characterisation and Engineering Properties of Natural Soils, Lisse: A.A. Balkema, 395–427. |
[3] |
Berre T, Lunne T, Andersen KH, et al. (2007) Potential improvements of design parameters by taking block samples of soft marine Norwegian clays. Can Geotech J 44: 698–716. doi: 10.1139/t07-011
![]() |
[4] | Berre T (2013) Test fill on soft plastic marine clay at Onsøy, Norway. Can Geotech J 51: 30–50. |
[5] |
Hight DW, Bond AJ, Legge JD (1992) Characterization of the Bothkennaar clay: an overview. Géotechnique 42: 303–347. doi: 10.1680/geot.1992.42.2.303
![]() |
[6] |
Ricceri G, Butterfield R (1974) An analysis of compressibility data from a deep borehole in Venice. Géotechnique 24: 175–192. doi: 10.1680/geot.1974.24.2.175
![]() |
[7] |
Cola S, Simonini P (2002) Mechanical behavior of silty soils of the Venice lagoon as a function of their grading characteristics. Can Geotech J 39: 879–893. doi: 10.1139/t02-037
![]() |
[8] |
Low HE, Maynard ML, Randolph MF, et al. (2011) Geotechnical characterisation and engineering properties of Burswood clay. Géotechnique 61: 575–591. doi: 10.1680/geot.9.P.035
![]() |
[9] |
Pineda JA, Suwal LP, Kelly RB, et al. (2016) Characterisation of Ballina clay. Géotechnique 66: 556–577. doi: 10.1680/jgeot.15.P.181
![]() |
[10] |
Kelly RB, Pineda JA, Bates L, et al. (2017) Site characterisation for the Ballina field testing facility. Géotechnique 67: 279–300. doi: 10.1680/jgeot.15.P.211
![]() |
[11] | DeGroot DJ, Lutenegger AJ (2003) Geology and engineering properties of Connecticut Valley Varved Clay. In: Tan TS, Phoon KK, Hight DW et al., editors. Characterisation and Engineering Properties of Natural Soils. Lisse: A.A. Balkema, 695–724. |
[12] |
Lutenegger AJ, Miller GA (1994) Uplift Capacity of Small-Diameter Drilled Shafts from In Situ Tests. J Geotech Eng 120: 1362–1380. doi: 10.1061/(ASCE)0733-9410(1994)120:8(1362)
![]() |
[13] |
Briaud JL, Gibbens R (1999) Behavior of Five Large Spread Footings in Sand. J Geotech Geoenvironmental Eng 125: 787–796. doi: 10.1061/(ASCE)1090-0241(1999)125:9(787)
![]() |
[14] | ISO (2002) Geotechnical investigation and testing-Identification and classification of soil. Part 1: Identification and description, Geneva, Switzerland: International Organization for Standardization. |
[15] | Norwegian Geotechnical Society (1989) Melding 7: Veiledning for utførelse av dreietrykksondering. Rev.1 [in Norwegian], Oslo, Norway: Norwegian Geotechnical Society (NGF). |
[16] | ISO (2012) Geotechnical investigation and testing-Field testing. Part 1: Electrical cone and piezocone penetration test, Geneva, Switzerland: International Organization for Standardization. |
[17] | ISO (2017) Geotechnical investigation and testing-Field testing. Part 11: Flat dilatometer test, Geneva, Switzerland: International Organization for Standardization. |
[18] | ISO (2012) Geotechnical investigation and testing-Field testing. Part 5: Flexible dilatometer test, Geneva, Switzerland: International Organization for Standardization. |
[19] | Norwegian Geotechnical Society (2017) Melding 6: Veiledning for måling av grunnvannsstand og poretrykk. Rev. 2 [In Norwegian], Oslo, Norway: Norwegian Geotechnical Society (NGF). |
[20] | Norwegian Geotechnical Society (1989) Melding 4: Veiledning for utførelse av vingeboring. Rev. 1 [in Norwegian], Oslo, Norway: Norwegian Geotechnical Society (NGF). |
[21] | Bjerrum L, Andersen KH (1972) In-situ measurements of lateral pressures in clay. European Conference on Soil Mechanics and Foundation Engineering, 5 Madrid 1972 Proceedings. Madrid: Sociedad Española de Mecánica del Suelo y Cimentaciones. |
[22] | Norwegian Geotechnical Society (2013) Melding 11: Veiledning for prøvetaking [In Norwegian], Oslo, Norway: Norwegian Geotechnical Society (NGF). |
[23] |
Lefebvre G, Poulin C (1979) A new method of sampling in sensitive clay. Can Geotech J 16: 226–233. doi: 10.1139/t79-019
![]() |
[24] |
Emdal A, Gylland A, Amundsen HA, et al. (2016) Mini-block sampler. Can Geotech J 53: 1235–1245. doi: 10.1139/cgj-2015-0628
![]() |
[25] | Huang AB, Tai YY, Lee WF, et al. (2008) Sampling and field characterization of the silty sand in Central and Southern Taiwan. In: Huang AB, Mayne PW, editors. Geotechnical and Geophysical Site Characterization. Leiden: Taylor & Francis, 1457–1463. |
[26] | Kazuo T, Kaneko S (2006) Undisturbed sampling method using thick water-soluble polymer solution Tsuchi-to-Kiso. J Jpn Geotech Soc 54: 145–148. |
[27] | ISO (2014) Geotechnical investigation and testing-Laboratory testing of soil. Part 1: Determination of water conten, Geneva, Switzerland: International Organization for Standardization. |
[28] | ISO (2014) Geotechnical investigation and testing-Laboratory testing of soil. Part 2: Determination of bulk density, Geneva, Switzerland: International Organization for Standardization. |
[29] | ISO (2015) Geotechnical investigation and testing-Laboratory testing of soil. Part 3: Determination of particle density, Geneva, Switzerland: International Organization for Standardization. |
[30] | ISO (2018) Geotechnical investigation and testing-Laboratory testing of soil. Part 12: Determination of liquid and plastic limits, Geneva, Switzerland: International Organization for Standardization. |
[31] |
Moum J (1965) Falling drop used for grain-size analysis of fine-grained materials. Sedimentology 5: 343–347. doi: 10.1111/j.1365-3091.1965.tb01566.x
![]() |
[32] | ISO (2016) Geotechnical investigation and testing-Laboratory testing of soil. Part 4: Determination of particle size distribution, Geneva, Switzerland: International Organization for Standardization. |
[33] | NS (1988) Geotechnical testing-Laboratory methods. Determination of undrained shear strength by fall-cone testing, Oslo, Norway: Standards Norway. |
[34] | ISO (1994) Soil quality. Determination of the specific electrical conductivity, Geneva, Switzerland: International Organization for Standardization. |
[35] | ISO (2017) Geotechnical investigation and testing-Laboratory testing of soil. Part 5: Incremental loading oedometer test, Geneva, Switzerland: International Organization for Standardization. |
[36] | Sandbækken G, Berre T, Lacasse S (1986) Oedometer Testing at the Norwegian Geotechnical Institute, In: Yong RN, Townsend FC, editors, Consolidation of soils: testing and evaluation, STP 892: American Society for Testing and Materials, 329–353. |
[37] | NS (1993) Geotechnical testing-Laboratory methods. Determination of one-dimensional consolidation properties by oedometer testing-Method using continuous loading, Oslo, Norway: Standards Norway. |
[38] | ISO (2004) Geotechnical investigation and testing-Laboratory testing of soil. Part 11: Determination of permeability by constant and falling head, Geneva, Switzerland: International Organization for Standardization. |
[39] |
Wang Z, Gelius LJ, Kong FN (2009) Simultaneous core sample measurements of elastic properties and resistivity at reservoir conditions employing a modified triaxial cell-a feasibility study. Geophy Prospec 57: 1009–1026. doi: 10.1111/j.1365-2478.2009.00792.x
![]() |
[40] |
Berre T (1982) Triaxial Testing at the Norwegian Geotechnical Institute. Geotech Test J 5: 3–17. doi: 10.1520/GTJ10794J
![]() |
[41] | ISO (2018) Geotechnical investigation and testing-Laboratory testing of soil. Part 9: Consolidated triaxial compression tests on water saturated soils, Geneva, Switzerland: International Organization for Standardization. |
[42] |
Bjerrum L, Landva A (1966) Direct Simple-Shear Tests on a Norwegian Quick Clay. Géotechnique 16: 1–20. doi: 10.1680/geot.1966.16.1.1
![]() |
[43] | ASTM (2015) Standard Test Method for Consolidated Undrained Direct Simple Shear Testing of Fine Grain Soils, West Conshohocken, PA: ASTM International. |
[44] | Dyvik R, Madshus C (1985) Lab measurements of Gmax using bender elements. In: Khosla V, editor. Advances in the Art of Testing Soils under Cyclic Conditions: Proceedings of a Session in Conjunction with the ASCE Convention in Detroit, Michigan 1985, New York: American Society of Civil Engineers, 186–196. |
[45] | Sørensen R (1999) En 14C datert og dendrokronologisk kalibrert strandforskyvningskurve for søndre Østfold, Sørøst-Norge [In Norwegian], In: Selsing L, Lillehammer G, editors, Museumslandskap: artikkelsamling til Kerstin Griffin på 60-årsdagen. Stavanger: Arkeologisk museum i Stavanger, 59–70. |
[46] | Klemsdal T (2002) Landformene i Østfold [In Norwegian]. Natur i Østfold 21: 7–31. |
[47] | Olsen L, Sørensen E (1993) Halden 1913 II, Quaternary map, 1:50.000, with descriptions, Trondheim: Geological Survey of Norway. |
[48] | Sørensen R (1979) Late Weichselian deglaciation in the Oslofjord area, south Norway. Boreas 8: 241–246. |
[49] |
Kenney TC (1964) Sea-Level Movements and the Geologic Histories of the Post-Glacial Marine Soils at Boston, Nicolet, Ottawa and Oslo. Géotechnique 14: 203–230. doi: 10.1680/geot.1964.14.3.203
![]() |
[50] |
Ostendorf DW, DeGroot DJ, Shelburne WM, et al. (2004) Hydraulic head in a clayey sand till over multiple timescales. Can Geotech J 41: 89–105. doi: 10.1139/t03-074
![]() |
[51] | Norwegian Geotechnical Society (2011) Melding 2: Veiledning for symboler og definisjoner i geoteknikk-Identifisering og klassifisering av jord Rev. 2 [In Norwegian], Oslo, Norway: Norwegian Geotechnical Society (NGF). |
[52] | Pettijohn FJ (1949) Sedimentary Rocks. New York: Harper and Row. |
[53] |
Rosenqvist IT (1975) Origin and Mineralogy Glacial and Interglacial Clays of Southern Norway. Clays Clay Miner 23: 153–159. doi: 10.1346/CCMN.1975.0230211
![]() |
[54] |
Solberg IL, Hansen L, Rønning JS, et al. (2012) Combined geophysical and geotechnical approach to ground investigations and hazard zonation of a quick clay area, mid Norway. Bull Eng Geol Environ 71: 119–133. doi: 10.1007/s10064-011-0363-x
![]() |
[55] |
Solberg IL, Rønning JS, Dalsegg E, et al. (2008) Resistivity measurements as a tool for outlining quick-clay extent and valley-fill stratigraphy: a feasibility study from Buvika, central Norway. Can Geotech J 45: 210–225. doi: 10.1139/T07-089
![]() |
[56] |
Hansen L, L'heureux JS, Longva O (2011) Turbiditic, clay-rich event beds in fjord-marine deposits caused by landslides in emerging clay deposits-palaeoenvironmental interpretation and role for submarine mass-wasting. Sedimentology 58: 890–915. doi: 10.1111/j.1365-3091.2010.01188.x
![]() |
[57] | Norwegian Geotechnical Institute (2002) Early Soil Investigations for Fast Track Projects: Assessment of Soil Design Parameters from Index Measurements in Clays, Summary Report/Manual 521553-3, Oslo: Norwegian Geotechnical Institute. |
[58] | Norwegian Geotechnical Society (2010) Melding 5: Veiledning for utførelse av trykksondering. Rev. 3 [in Norwegian], Oslo, Norway: Norwegian Geotechnical Society (NGF). |
[59] | Lunne T, Strandvik SO, Kåsin K, et al. (2018) Effect of cone penetrometer type on CPTU results at a soft clay test site in Norway. In: Hicks MA, Pisanò F, Peuchen J, editors, Cone Penetration Testing 2018. Leiden: CRC Press, 417–422. |
[60] |
Robertson PK (1990) Soil classification using the cone penetration test. Can Geotech J 27: 151–158. doi: 10.1139/t90-014
![]() |
[61] | Marchetti S (1980) In Situ Tests by Flat Dilatometer. J Geotech Eng Div 106: 299–321. |
[62] | Marchetti S, Monaco P, Totani G, et al. (2006) The Flat Dilatometer Test (DMT) in soil investigations A Report by the ISSMGE Committee TC16. In: Failmezger RA, Anderson JB, editors. Proceedings from the Second International Conference on the Flat Dilatometer, Washington, DC, April 2–5, 2006, Lancaster, VA: In-Situ Soil Testing. |
[63] |
Marsland A, Randolph MF (1977) Comparisons of the results from pressuremeter tests and large in situ plate tests in London Clay. Géotechnique 27: 217–243. doi: 10.1680/geot.1977.27.2.217
![]() |
[64] | Casagrande A (1936) The determination of the preconsolidation load and its practical significance. Proceedings of the First International Conference on Soil Mechanics and Foundation Engineering: Cambridge, MA, 22–26 June 1936. Cambridge: Graduate school of engineering, Harvard University, 60–64. |
[65] | Janbu N (1963) Soil compressibility as determined by oedometer and triaxial tests. Problems of Settlements and Compressibility of Soils: Proceedings: European Conference of Soil Mechanics and Foundation Engineering, Wiesbaden, Germany, 19–25. |
[66] | Pacheco Silva F (1970) A new graphical construction for determination of the pre-consolidation stress of a soil sample. Proceedings of the 4th Brazilian Conference on Soil Mechanics and Foundation Engineering, Rio de Janeiro, Brazil, 225–232. |
[67] | Lunne T, Robertson PK, Powell JJM (1997) Cone penetration testing in geotechnical practice. London: Blackie Academic & Professional. |
[68] | Mayne PW (2007) Cone Penetration Testing: A Synthesis of Highway Practice. NCHRP Synthesis 368. Washington, D.C.: Transportation Research Board. |
[69] | Chandler RJ (1988) The In-Situ Measurement of the Undrained Shear Strength of Clays Using the Field Vane. In: Richards AF, editor. Vane Shear Strength Testing in Soils: Field and Laboratory Studies, STP 1014. West Conshohocken, PA: ASTM International, 13–32. |
[70] |
Mesri G, Hayat TM (1993) The coefficient of earth pressure at rest. Can Geotech J 30: 647–666. doi: 10.1139/t93-056
![]() |
[71] |
Palmer AC (1972) Undrained plane-strain expansion of a cylindrical cavity in clay: a simple interpretation of the pressuremeter test. Géotechnique 22: 451–457. doi: 10.1680/geot.1972.22.3.451
![]() |
[72] |
Wroth CP (1984) The interpretation of in situ soil tests. Géotechnique 34: 449–489. doi: 10.1680/geot.1984.34.4.449
![]() |
[73] | Rix GJ, Stoke KH (1991) Correlation of Initial Tangent Modulus and Cone Resistance. In: Huang AB, editor. Calibration Chamber Testing: Proceedings of the First International Symposium (ISOCCTI), Potsdam, NY, USA, 28-29 June, 1991: Elsevier, 351–362. |
[74] |
Mayne PW, Rix GJ (1995) Correlations between shear wave velocity and cone tip resistance in natural clays. Soils Found 35: 107–110. doi: 10.3208/sandf1972.35.2_107
![]() |
[75] |
Janbu N (1985) Soil models in offshore engineering. Géotechnique 35: 241–281. doi: 10.1680/geot.1985.35.3.241
![]() |
[76] |
Carroll R, Long M (2017) Sample Disturbance Effects in Silt. J Geotech Geoenvironmental Eng 143: 04017061. doi: 10.1061/(ASCE)GT.1943-5606.0001749
![]() |
[77] |
Martins FB, Bressani LA, Coop MR, et al. (2001) Some aspects of the compressibility behaviour of a clayey sand. Can Geotech J 38: 1177–1186. doi: 10.1139/t01-048
![]() |
[78] |
Long M (2007) Engineering characterization of estuarine silts. Q J Eng Geol Hydrogeol 40: 147–161. doi: 10.1144/1470-9236/05-061
![]() |
[79] |
Long M, Gudjonsson G, Donohue S, et al. (2010) Engineering characterisation of Norwegian glaciomarine silt. Eng Geol 110: 51–65. doi: 10.1016/j.enggeo.2009.11.002
![]() |
[80] | Skúlasson J (1996) Settlement investigation on Icelandic silt. In: Erlingsson S, Sigursteinsson H, editors. Interplay between geotechnics and environment : XII Nordic Geotechnical Conference, NGM-96, Reykjavik, 1996. Reykjavik: Jardtæknifélag Islands, 435–441. |
[81] | DeJong JT, Jaeger RA, Boulanger RW, et al. (2013) Variable penetration rate cone testing for characterization of intermediate soils. In: Coutinho RQ, Mayne PW, editors. Geotechnical and Geophysical Site Characterization 4, Boca Raton, FL: Taylor & Francis, 25–42. |
[82] | Taylor DW (1948) Fundamentals of soil mechanics. New York: J. Wiley. |
[83] | Ladd CC, Weaver JS, Germaine JT, et al. (1985) Strength-Deformation Properties of Arctic Silt. In: F. Lawrence Bennett, Jerry L. Machemehl, Thelen NDW, editors. Civil Engineering in the Arctic Offshore: Conference Arctic '85, San Francisco, CA, March 25-27, 1985. New York: American Society of Civil Engineers, 820–829. |
[84] | Sandven R (2003) Geotechnical properties of a natural silt deposit obtained from field and laboratory tests. In: Tan TS, Phoon KK, Hight DW et al., editors. Characterisation and Engineering Properties of Natural Soils. Lisse: A.A. Balkema, 1121–1148. |
[85] | Carroll R, Paniagua López AP (2018) Variable rate of penetration and dissipation test results in a natural silty soil. In: Hicks MA, Pisanò F, Peuchen J, editors. Cone Penetration Testing 2018: Proceedings of the 4th International Symposium on Cone Penetration Testing (CPT'18), 21–22 June, 2018, Delft, The Netherlands, London: CRC Press. |
[86] |
Sully JP, Robertson PK, Campanella RG, et al. (1999) An approach to evaluation of field CPTU dissipation data in overconsolidated fine-grained soils. Can Geotech J 36: 369–381. doi: 10.1139/t98-105
![]() |
[87] | Larsson R (1997) Investigations and load tests in silty soils. Results from a series of investigations in silty soils in Sweden. Report 54. Linköping: Swedish Geotechnical Institute, SGI, 257. |
[88] |
Blight GE (1968) A Note on Field Vane Testing of Silty Soils. Can Geotech J 5: 142–149. doi: 10.1139/t68-014
![]() |
[89] | Gibson RE, Anderson WF (1961) In situ measurement of soil properties with the pressuremeter Civ Engi Public Work Rev 56: 615–618. |
[90] |
Aubeny CP, Whittle AJ, Ladd CC (2000) Effects of Disturbance on Undrained Strengths Interpreted from Pressuremeter Tests. J Geotech Geoenvironmental Eng 126: 1133–1144. doi: 10.1061/(ASCE)1090-0241(2000)126:12(1133)
![]() |
[91] | Senneset K, Sandven R, Lunne T, et al. (1988) Piezocone tests in silty soils. In: de Ruiter J, editor. Penetration testing, 1988: proceedings of the First International Symposium on Penetration Testing, ISOPT-1, Orlando, 20-24 March 1988. Rotterdam, The Netherlands: A.A. Balkema, 863–870. |
[92] |
Brandon TL, Rose AT, Duncan JM (2006) Drained and undrained strength interpretation for low-plasticity silts. J Geotech Geoenvironmental Eng 132: 250–257. doi: 10.1061/(ASCE)1090-0241(2006)132:2(250)
![]() |
[93] |
Robertson PK, Campanella RG (1983) Interpretation of cone penetration tests. Part I: Sand. Can Geotech J 20: 734–745. doi: 10.1139/t83-079
![]() |
[94] | Kulhawy FH, Mayne PW (1990) Manual on Estimating Soil Properties for Foundation Design, Report EL-6800, Electric Power Research Institute, Palo Alto, CA. |
[95] | Janbu N, Senneset K (1974) Effective stress interpretation of in-situ static penetration tests. Proceedings of the European Symposium on Penetration Testing, ESOPT, Stockholm, June 5-7, 1974. Stockholm: National Swedish Building Research, 181–193. |
[96] | Senneset K, Sandven R, Janbu N (1989) Evaluation of soil parameters from piezocone tests. Transportation Research Record 1235: 24–37. |
[97] | Börgesson L (1981) Shear strength of inorganic silty soils. Proceedings of the 10th International Conference on Soil Mechanics and Foundation Engineering: 15–19 June, Stockholm,1981. Rotterdam: A.A. Balkema, 567–572. |
[98] |
Høeg K, Dyvik R, Sandbækken G (2000) Strength of undisturbed versus reconstituted silt and silty sand specimens. J Geotech Geoenvironmental Eng 126: 606–617. doi: 10.1061/(ASCE)1090-0241(2000)126:7(606)
![]() |
[99] | Terzaghi K, Peck RB, Mesri G (1996) Soil Mechanics in Engineering Practice: John Wiley and Sons. |
[100] | Lunne T, Berre T, Strandvik S (1997) Sample disturbance effects in soft low plastic Norwegian clay. In: Almeida M, editor. Recent Developments in Soil and Pavement Mechanics: Proceedings of the International Symposium, Rio de Janeiro, Brazil, 25–27 June 1997. Rotterdam: A.A. Balkema, 81–102. |
[101] | Hight DW, Leroueil S (2003) Characterisation of soils for engineering purposes,. In: Tan TS, Phoon KK, Hight DW et al., editors. Characterisation and Engineering Properties of Natural Soils. Lisse: A.A. Balkema, 255–360. |
[102] | Solhjell E, Strandvik SO, Carroll R, et al. (2017) Johan Sverdrup–Assessment of soil material behaviour and strength properties for the shallow silt layer. Offshore Site Investigation and Geotechnics, Smarter Solutions for Future Offshore Developments: Proceedings of the 8th International Conference 12-14 September 2017 Royal Geographical Society, London, UK, London: Society for Underwater Technology, 1275–1282. |
[103] |
Bray JD, Sancio RB, Durgunoglu T, et al. (2004) Subsurface Characterization at Ground Failure Sites in Adapazari, Turkey. J Geotech Geoenvironmental Eng 130: 673–685. doi: 10.1061/(ASCE)1090-0241(2004)130:7(673)
![]() |
[104] | Arroyo M, Pineda JA, Sau N, et al. (2015) Sample quality examination in silty soils. In: Winter MG, Smith DM, Eldred PJL et al., editors. Geotechnical engineering for infrastructure and development: proceedings of the XVI European Conference on Soil Mechanics and Geotechnical Engineering, London: ICE Publishing, 2873–2878. |
[105] | Bradshaw AS, Baxter CDP (2007) Sample Preparation of Silts for Liquefaction Testing. Geotech Test J 30: 324–332. |
[106] | Sau N, Arroyo M, Pérez N, et al. (2014) Using CAT to obtain density maps in Sherbrooke specimens of silty soils. In: Soga K, Kumar K, Biscontin G et al., editors. Geomechanics from micro to macro: proceedings of the TC105 ISSMGE International Symposium on Geomechanics from Micro to Macro, Cambridge, UK, 1-3 September 2014. Leiden, Netherlands: CRC Press, 1153–1158. |
[107] |
LaRochelle P, Sarrailh J, Tavenas F, et al. (1981) Causes of sampling disturbance and design of a new sampler for sensitive soils. Can Geotech Journal 18: 52–66. doi: 10.1139/t81-006
![]() |
[108] |
DeJong JT, Randolph M (2012) Influence of Partial Consolidation during Cone Penetration on Estimated Soil Behavior Type and Pore Pressure Dissipation Measurements. J Geotech Geoenvironmental Eng 138: 777–788. doi: 10.1061/(ASCE)GT.1943-5606.0000646
![]() |
[109] |
Vesterberg B, Bertilsson R, Löfroth H (2017) Photographic feature: Monitoring of negative porewater pressure in silt slopes. Q J Eng Geol Hydrogeol 50: 245–248. doi: 10.1144/qjegh2016-083
![]() |
[110] | Westerberg B, Bertilsson R, Prästings A, et al. (2014) Publication 9: Negativa portryck och stabilitet i siltslänter. Linköping: Statens Geotekniska Institut, SGI [in Swedish, with summary in English]. |
[111] | Clausen CJF (2003) BEAST: A computer program for limit equilibrium analysis by method of slices, Report 8302-2. Rev. 4, 24 April. |
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