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Investigating minimal requirements for plants on textile substrates in low-cost hydroponic systems

  • With a growing world population and the concentration of citizens in big cities new methods of agriculture are required. Vertical farming attracts more attention in mending these growing problems. To enable a widespread use of low-cost hydroponic systems this study investigates minimal requirements for plants (different herbs and vegetables) in such a hydroponic vertical farming system and the suitability of textiles as sustainable substrates. Therefore, this study aims to investigate plant stress levels, germination rates and water usage in a low-cost hydroponic system with no special lightning in principle comparison with indoor cultivation in soil. The results of the pulse-amplitude-modulation (PAM) measurements as measure of photosynthetic performance indicate that the plants were equally stressed in hydroponic and in soil cultivation. In this respect, the photosynthetic quantum yield in both cultivation systems is on average only slightly lower than the values expected under optimal conditions. It was observed that chive and lovage not only had a significantly higher germination rate in the hydroponic system but also accumulated significantly more fresh as well as dry biomass, while spinach, thyme and marjoram showed higher germination rates in soil cultivation. The water consumption in the setup was considerably higher for the hydroponic system compared to indoor soil cultivation.

    Citation: Bennet Brockhagen, Fabian Schoden, Jan Lukas Storck, Timo Grothe, Christian Eßelmann, Robin Böttjer, Anke Rattenholl, Frank Gudermann. Investigating minimal requirements for plants on textile substrates in low-cost hydroponic systems[J]. AIMS Bioengineering, 2021, 8(2): 173-191. doi: 10.3934/bioeng.2021016

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  • With a growing world population and the concentration of citizens in big cities new methods of agriculture are required. Vertical farming attracts more attention in mending these growing problems. To enable a widespread use of low-cost hydroponic systems this study investigates minimal requirements for plants (different herbs and vegetables) in such a hydroponic vertical farming system and the suitability of textiles as sustainable substrates. Therefore, this study aims to investigate plant stress levels, germination rates and water usage in a low-cost hydroponic system with no special lightning in principle comparison with indoor cultivation in soil. The results of the pulse-amplitude-modulation (PAM) measurements as measure of photosynthetic performance indicate that the plants were equally stressed in hydroponic and in soil cultivation. In this respect, the photosynthetic quantum yield in both cultivation systems is on average only slightly lower than the values expected under optimal conditions. It was observed that chive and lovage not only had a significantly higher germination rate in the hydroponic system but also accumulated significantly more fresh as well as dry biomass, while spinach, thyme and marjoram showed higher germination rates in soil cultivation. The water consumption in the setup was considerably higher for the hydroponic system compared to indoor soil cultivation.



    Throughout this paper, the following notations are used. R, RBQ represent the set of real number and reduced biquaternion, respctively. Rt(Rt) represent the set of all real column(row) vectors with order t. Rm×n, RBm×nQ represent the set of all m×n real matrices, reduced biquaternion matrices, respectively. RBn×nHQ, RBn×nAQ represent the set of all n×n Hermitian reduced biquaternion matrices and Anti-Hermitian reduced biquaternion matrices, respectively. Ik represents the unit matrix with order k, δik represents the ith column of unit matrix Ik. δk[i1, , is] is a abbreviation of [δi1k, , δisk]. A=A1+A2i+A3j+A4kRBm×nQ, where AiRm×n, (i=1:4) and define ¯A=A1A2iA3jA4k to be conjugate of A. AT, AH, A represent the transpose, conjugate transpose, Moore-Penrose(MP) inverse of matrix A. represents the Kronecker product of matrices. represents the semi-tensor product of matrices. represents the Frobenius norm of a matrix or Euclidean norm of a vector.

    The concept of quaternion was proposed by Hamilton in 1843, which is an extension of complex number. It consists of four parts, i.e.

    a=ar+aii+ajj+akk,

    where ar, ai, aj, ak are real numbers and i, j, k satisfy

    i2=j2=k2=1, ij=ji=k, jk=kj=i, ki=ik=j.

    As can be seen from the product rule above, quaternion is not commutative with respect to multiplication. Owing to this reason, quaternion becomes more complicated in some operations, and it will have great trouble in application to practical problems. This also promotes the generation of reduced biquaternion to some extent. Reduced biquaternion is similar to quaternion in form, but it has a different product rule.

    Definition of reduced biquaternion [1]

    a=ar+aii+ajj+akk,

    where ar, ai, aj, ak are real numbers and

    i2=k2=1, j2=1, ij=ji=k, jk=kj=i, ki=ik=j.

    e1=1+j2, e2=1j2 are two special numbers in reduced biquaternion. Obviously,

    en1=en11==e1, en2=en12==e2,e1e2=0.

    Therefore, e1 and e2 are both idempotent elements and divisors of zero. Any reduced biquaternion with the form k1e1 or k2e2 is also a divisor of zero and does not have a multiplicative inverse, where, k1, k2 are arbitrary complex numbers. By means of e1 and e2, we can uniquely express the reduced biquaternion a=ar+aii+ajj+akk as a=a1e1+a2e2, where a1=(ar+aj)+(ai+ak)i, a2=(araj)+(aiak)i.

    The conjugate of a reduced biquaternion a is denoted by ¯a and ¯a=araiiajjakk. The norm of a reduced biquaternion a is

    a=a2r+a2i+a2j+a2k.

    Then, the Frobenius norm of ARBm×nQ is defined as follows

    A=mi=1nj=1aij2.

    The concept of reduced biquaternion was first defined by Sch¨utte and Wenzel in 1990 [2]. It can be seen that the product of reduced biquaternions has commutability compared with quaternion. Thus, many operations of reduced biquaternion are simpler than those of quaternion. Such as, the implementations of the discrete reduced biquaternion Fourier transform, convolution, correlation. Therefore, it is of importance to study the theoretical knowledge and numerical calculation of reduced biquaternions. Many good results on reduced biquaternions have obtained. For example, Pei et al. investigated digital signal and image processing using reduced biquaternion in 2004 [1] and gave the algorithms for calculating the eigenvalues, the eigenvectors, and the singular value decomposition of a reduced biquaternion matrix in 2008 [3]; Isokawa et al. studied two types of multistate Hopfield neural networks using reduced biquaternion in 2010 [4]. At the same time, scholars also focused on the solution of specific reduced biquaternion matrix equations because matrix equations have a wide range of applications in control theory, stability and other fields. Hidayet derived the expressions of the minimal norm least squares solution for the reduced biquaternion matrix equation AX=B using the e1e2 form in 2019 [5]; Yuan studied the Hermitian solutions of the reduced biquaternion matrix equation (AXB,CXD)=(E,F) using complex representation [6].

    In this paper, we study the least squares problems of the reduced biquaternion matrix equation

    ki=1AiXBi=C. (1.1)

    Since Hermitian and Anti-Hermitian matrices are very useful in engineering problems and linear system theory. So many researchers turn on the problems of (Anti)-Hermitian matrix equations, for example, [7,8] derived explicit determinantal representation formulas of the general, Hermitian, and Anti-Hermitian solutions to the system of two-sided quaternion matrix equations A1XA1=C1 and A2XA2=C2 in 2018 and the Sylvester type matrix equation AXA+BYB=C in 2019 using determinantal representations of Moore-Penrose inverse, respectively. [9] proposed a recursive algorithm for calculating the inversion of the confluent Vandermonde matrix that with consecutive powers devoted just to the Hermite type interpolation and derived an explicit analytic formula for the calculation of the inverse of the confluent Vandermonde matrix in 2013. [10] considered the quaternion matrix equation XAˆXB=C and studied its minimal norm least squares solution, j-self-conjugate least squares solution and anti-j-self-conjugate least squares solution by means of real representation matrices of quaternion matrix in 2020.

    We determine our research objective as the least squares Hermitian solution and the least squares Anti-Hermitian solution as follows:

    Problem 1. Let AiRBm×nQ, BiRBp×qQ, CRBm×qQ, and

    SQ={X|XRBn×pQ,  ki=1AiXBi=C}.

    Find out XQSQ such that

    XQ=minXSQX.

    Problem 2. Let AiRBm×nQ, BiRBn×qQ, CRBm×qQ, and

    SHQ={X|XRBn×nHQ,  ki=1AiXBi=C}.

    Find out XHQSHQ such that

    XHQ=minXSHQX.

    Problem 3. Let AiRBm×nQ, BiRBn×qQ, CRBm×qQ, and

    SAQ={X|XRBn×nAQ,  ki=1AiXBi=C}.

    Find out XAQSAQ such that

    XAQ=minXSAQX.

    The semi-tensor product(STP) of matrices was proposed initially by Cheng to solve linearization problem of nonlinear systems, which is a generalization of traditional matrix product for the case when the two factor matrices do not meet the dimension matching condition. It has been proved to be a power tool in many fields such as game theory [11], graph coloring [12], logic systems [13] and so on. In this paper, we will convert the least squares problem of reduced biquaternion matrix equation to the corresponding real problems by using the semi-tensor product of matrices.

    This paper is organized as follows. In Section 2, some basic knowledge of semi-tensor product of matrices is introduced. In Section 3, a new kind of real vector representation of a reduced biquaternion matrix and the main properties are proposed. In Section 4, the solutions of Problems 1–3 are studied by using the real vector representation of reduced biquaternion matrix, the special structure of solutions and semi-tensor product of matrices. In Section 5, two examples are illustrated to demonstrate the Algorithms. Finally the present thesis is summarized.

    In this section, we will recall some basic knowledge of semi-tensor product of matrices. Please refer to [14,15] for more details.

    Definition 2.1. Let ARm×n, BRp×q, the semi-tensor product of A and B, denoted by

    AB=(AIt/n)(BIt/p),

    where t=lcm(n,p) is the least common multiple of n and p. If n=p, the semi-tensor product reduces to the traditional matrix product.

    Next, a simple numerical example is used to explain the semi-tensor product of matrices.

    Example 2.1. Suppose A=[221110332321], B=[1221], then

    AB=A(BI2)=[221110332321][1020010220100201]=[445556136567].

    Lemma 2.1. Assume that A, B, C are real matrices with appropriate sizes, a, bR.

    (1) (Distributive law) A(aB±bC)=aAB±bAC, (aA±bB)C=aAC±bBC.

    (2) (Associative law)(AB)C=A(BC).

    (3) (Transpose)(AB)T=BTAT.

    (4) (Inverse)(AB)1=B1A1, where A and B are invertible square matrices.

    (5) Assume that xRm, yRn, then xy=xy.

    It can be seen from the Definition 2.1 that semi-tensor product of matrices is a generalization of traditional matrix product. A large part of the properties of traditional matrix product are preserved by semi-tensor product of matrices. Here, only (3) of Lemma 2.1 is simply proved and the other properties of Lemma 2.1 are similarly proved.

    Proof. Suppose ARm×n, BRp×q, t=lcm(n,p), we can obtain

    (AB)T=((AIt/n)(BIt/p))T=(BIt/p)T(AIt/n)T=(BTIt/p)(AT×It/n)=BTAT.

    The semi-tensor product of a matrix and a vector has the following property of quasi-commutativity.

    Definition 2.2. A swap matrix W[m,n] is a mn×mn matrix, which is defined as

    W[m,n]=δmn[1, , (n1)m+1, , m, , nm]Rmn×mn.

    We use the following example to briefly illustrate the construction of swap matrix.

    Example 2.2. Suppose m=2, n=3, then

    W[m,n]=δ6[1,3,5,2,4,6]=[δ16,δ36,δ56,δ26,δ46,δ66]=[100000000100010000000010001000000001]

    Remark 2.1. When m=n, we denote W[m,n]=W[n].

    The function of a swap matrix is to exchange the order of two vectors in vector multiplication.

    Lemma 2.2. Let xRm and yRn be two column vectors. Then W[m,n]xy=yx.

    Example 2.3. Suppose x=[x1x2], y=[y1y2y3]. Using the swap matrix constructed in Example 2.2, we can obtain

    W[m,n]xy=[100000000100010000000010001000000001]([x1x2][y1y2y3])=[100000000100010000000010001000000001][x1y1x1y2x1y3x2y1x2y2x2y3]=[y1x1y1x2y2x1y2x2y3x1y3x2]=([y1y2y3][x1x2])=yx

    Lemma 2.3. Assume ARm×n is given, xRt, ωRt. Then

    xA=(ItA)x,
    Aω=ω(ItA).

    Proof. Suppose x=[x1,x2,,xt]TRt, A=[a11a1nam1amn]Rm×n, thus

    xA=(xIm)A=[x1ImxtIm][a11a1nam1amn]=[x100xt000x100xt0000x1000xt]T[a11a1nam1amn]=[x1Ax2AxtA]=(ItA)x.

    When ω is a row vector, we use the transpose property of the semi-tensor product of matrices to derive the transformation.

    Definition 2.3. Let Wi (i=0,1,,n) be vector spaces. The mapping F:Πni=1WiW0 is called a multilinear mapping, if for any 1in, α, βR,

    F(x1,,xi1,αxi+βyi,,xn)=αF(x1,, xi,,xn)+βF(x1,,yi,, xn),

    in which xiWi,1in,yiWi. If dim(Wi)=ki, (i=0,1,, n), and (δ1ki, δ2ki, , δkiki) is the basis of Wi. Denote

    F(δj1k1, δj2k2, , δjnkn)=k0s=1 cj1, j2, , jnsδsk0,

    in which jt=1, , kt, t=1, , n. Then

    {cj1, j2, , jns| jt=1, , kt, t=1, , n, s=1, , k0}

    are called structure constants of F. Arranging these structure constants in the following form

    MF=[c1111c11kn1ck1k2kn1c1112c11kn2ck1k2kn2  c111k0c11knk0ck1k2knk0],

    MF is called the structure matrix of F.

    In this section, we will propose the concept of real vector representation of a reduced biquaternion matrix and study its properties.

    Definition 3.1. Let x=x1+x2i+x3j+x4kRBQ, denote vR(x)=[x1,x2,x3,x4]T, vR(x) is called as the real vector representation of x.

    Theorem 3.1. Let x,yRBQ, then

    vR(xy)=MQvR(x)vR(y), (3.1)

    where

    MQ=[1000010000100001010010000001001000100001100001000001001001001000],

    MQ is the structure matrix of multiplication of reduced biquaternions.

    Proof. Suppose x=x1+x2i+x3j+x4k, y=y1+y2i+y3j+y4k, we can obtain

    xy=(x1+x2i+x3j+x4k)(y1+y2i+y3j+y4k)=(x1y1x2y2+x3y3x4y4)+(x1y2+x2y1+x3y4+x4y3)i+(x1y3+x3y1x2y4x4y2)j+(x1y4+x4y1+x2y3+x3y2)k,

    thus the left hand side of (3.1) is

    [x1y1x2y2+x3y3x4y4x1y2+x2y1+x3y4+x4y3x1y3+x3y1x2y4x4y2x1y4+x4y1+x2y3+x3y2].

    Since the right hand side of (3.1) is

    MQvR(x)vR(y)=MQ[x1x2x3x4][y1y2y3y4]=MQ([x1x2x3x4][y1y2y3y4])=[x1y1x2y2+x3y3x4y4x1y2+x2y1+x3y4+x4y3x1y3+x3y1x2y4x4y2x1y4+x4y1+x2y3+x3y2].

    (3.1) can be obtained.

    Combined the real vector representation of a reduced biquaternion with vec operator of a real matrix, we propose a new kind of real vector representation of a reduced biquaternion matrix. For this purpose, we first propose the real vector representation of a reduced biquaternion vector as follows.

    Definition 3.2. Let x=[x1,,xn], y=[y1,,yn]T be reduced biquaternion vectors. Denote

    vR(x)=[vR(x1)vR(xn)],  vR(y)=[vR(y1)vR(yn)],

    vR(x) and vR(y) are called as the real vector representation of reduced biquaternion vectors x and y.

    Now we define the concepts of the real vector representation of a reduced biquaternion matrix A.

    Definition 3.3. For A=(Aed)RBm×nQ, e=1,,m, d=1,,n, denote

    vRc(A)=[vR(A11)vR(Am1)vR(A1n)vR(Amn)]=[vR(Col1(A))vR(Col2(A))vR(Coln(A))],  vRr(A)=[vR(A11)vR(A1n)vR(Am1)vR(Amn)]=[vR(Row1(A))vR(Row2(A))vR(Rowm(A))],

    vRc(A) and vRr(A) are called the real column stacking form and the real row stacking form of A, respectively. Real column stacking form and real row stacking form of A are collectively called real vector representation of A.

    We can prove that this real vector representation has the following properties with respect to vector or matrix operations.

    Theorem 3.2. Let x=[x1,x2,,xn], ˇx=[ˇx1,ˇx2,,ˇxn], y=[y1,y2,,yn]T, aR, xi, ˇxi, yiRBQ, then

    (1) vR(x+ˇx)=vR(x)+vR(ˇx),(2) vR(ax)=avR(x),(3) vR(xy)=MQni=1(δin)T(I4n(δin)T)vR(x)vR(y).

    Proof. By simply computing, we know (1), (2) hold. We only give a detailed proof of (3). Using Theorem 3.1, we have

    vR(xy)=vR(x1y1+x2y2++xnyn)=MQvR(x1)vR(y1)++MQvR(xn)vR(yn)=MQ(δ1n)TvR(x)(δ1n)TvR(y)++MQ(δnn)TvR(x)(δnn)TvR(y)=MQ(δ1n)T(I4n(δ1n)T)vR(x)vR(y)++MQ(δnn)T(I4n(δnn)T)vR(x)vR(y)=MQni=1(δin)T(I4n(δin)T)vR(x)vR(y).

    By using Theorem 3.2, we can drive the following results on the real vector representation of multiplication of two reduced biquaternion matrices.

    Theorem 3.3. Let A, ˇARBm×nQ, BRBn×pQ, αR, then

    (1) vRr(A+ˇA)=vRr(A)+vRr(ˇA),   vRc(A+ˇA)=vRc(A)+vRc(ˇA),(2) vRr(αA)=αvRr(A),   vRc(αA)=αvRc(A),(3) vRr(AB)=GvRr(A)vRc(B),   vRc(AB)=GvRr(A)vRc(B),

    in which

    G=[F(δ1m)T[I4mn(δ1p)T]F(δ1m)T[I4mn(δpp)T]F(δmm)T[I4mn(δ1p)T]F(δmm)T[I4mn(δpp)T]],  G=[F(δ1m)T[I4mn(δ1p)T]F(δmm)T[I4mn(δ1p)T]F(δ1m)T[I4mn(δpp)T]F(δmm)T[I4mn(δpp)T]],

    and F=MQni=1(δin)T(I4n(δin)T).

    Proof. We still only prove the first equality in (3). We block A and B with its rows or columns as follows

    A=[Row1(A)Row2(A)Rowm(A)], B=[Col1(B) Col2(B)  Colp(B)].

    Then we have

    vRr(AB)=[vR(Row1(A)Col1(B))vR(Row1(A)Colp(B))vR(Rowm(A)Col1(B))vR(Rowm(A)Colp(B))]=[FvR(Row1(A))vR(Col1(B))FvR(Row1(A))vR(Colp(B))FvR(Rowm(A))vR(Col1(B))FvR(Rowm(A))vR(Colp(B))]=[F[(δ1m)TvRr(A)][(δ1p)TvRc(B)]F[(δ1m)TvRr(A)][(δpp)TvRc(B)]F[(δmm)TvRr(A)][(δ1p)TvRc(B)]F[(δmm)TvRr(A)][(δpp)TvRc(B)]]=[F(δ1m)T[I4mn(δ1p)T]F(δ1m)T[I4mn(δpp)T]F(δmm)T[I4mn(δ1p)T]F(δmm)T[I4mn(δpp)T]]vRr(A)vRc(B).

    The second equality can be proved similarly.

    In this section, we study Problems 1–3. By means of the real vector representation of reduced biquaternion matrix and STP, we first convert Problems 1–3 into the corresponding real least squares problems. And then we obtain their solutions.

    Lemma 4.1. [16]The matrix equation Ax=b, with ARm×n and bRm, has a solution xRn if and only if

    AAb=b.

    In that case it has the general solution

    x=Ab+(InAA)y,

    where yRn is an arbitrary vector. The minimal norm solution of the linear system of equations Ax=b is Ab

    Theorem 4.2. Let AiRBm×nQ, BiRBp×qQ, CRBm×qQ, denote

    M=G1G(I4mnW[4pq,4np])(ki=1vRr(Ai)vRc(Bi)),

    where G1 has the same structure as G in Theorem 3.3 excepting the dimension. Hence the set SQ of Problem 1 is represented as

    SQ={XvRc(X)=MvRr(C)+(I4npMM)y,   yR4np}. (4.1)

    And then, the minimal norm solution XQ of Problem 1 satisfies

    vRc(XQ)=MvRr(C). (4.2)

    Proof.

    ki=1AiXBi=CvRr(ki=1AiXBi)=vRr(C)G1G(I4mnW[4pq,4np])(ki=1vRr(Ai)vRc(Bi))vRc(X)=vRr(C)MvRc(X)=vRr(C).

    For the real matrix equation

    MvRc(X)=vRr(C),

    by Lemma 4.1, its solutions can be represented as

    vRc(X)=MvRr(C)+(I4npMM)y,   yR4np.

    Thus we get the formula in (4.1).

    Notice

    minXRBn×pQXminvRc(X)R4npvRc(X)

    According to the previous proof of this theorem, we have that the minimal norm solution XQSQ of Problem 1 satisfies

    vRc(XQ)=MvRr(C).

    Therefore, (4.2) holds.

    By Theorem 4.2, we can get the sufficient and necessary condition of compatibility of the reduced biquaternion matrix equation ni=1AiXBi=C and the expression of the solution when ni=1AiXBi=C is compatible.

    Corollary 4.3. Let AiRBm×nQ, BiRBp×qQ, CRBm×qQ, M is given in Theorem 4.2. Then the following statements are equivalent:

    (a) Problem 1 has a solution XSQ;

    (b) (MMI4mq)vRr(C)=0.

    Moreover, if (b) holds, the solution set of (1.1) over RBn×pQ can be represented as

    SQ={XvRc(X)=MvRr(C)+(I4npMM)y,   yR4np}.

    Proof.

    ki=1AiXBi=CMvRc(X)=vRr(C)MMMvRc(X)=vRr(C)MMvRr(C)=vRr(C)(MMI4mq)vRr(C)=0.

    In order to study Problem 2, we define vRs(X) and give the relation of vRs(X) and vRc(X) for an Hermitian matrix.

    Theorem 4.4. Let X=[x11x1nxn1xnn]RBn×nHQ, xij=x1ij+x2iji+x3ijj+x4ijkRBQ, denote

    LXi=[xiivR(xi(i+1))vR(xin)],  vRs(X)=[LX1LX2LXn],  J=[J1JrJn],  Jr=[J1rJmrJnr],
    Jmrm<r={(δγ+η+22n2n)T(δγ+η+32n2n)T(δγ+η+42n2n)T(δγ+η+52n2n)T, Jmrm=r={(δλ+12n2n)T01×2n2n01×2n2n01×2n2n, Jmrm>r={(δλ+θ+22n2n)T(δλ+θ+32n2n)T(δλ+θ+42n2n)T(δλ+θ+52n2n)T,

    where, γ=(m1)(8n+24m)2, η=4(rm1), λ=(r1)(8n+24r)2, θ=4(mr1). Then

    vRc(X)=JvRs(X).

    Remark 4.1. It can be seen from the structural characteristics of matrix X that only part of the elements in matrix X can be used to represent the whole matrix X. So we need to find all the non-zero and non-repeating elements in X, which reduces the number of elements in vRc(X). J is a correspondence between the real vector representation of the matrix X and the real vector representation of the independent elements of the matrix X.

    Theorem 4.5. Let AiRBm×nQ, BiRBn×qQ, CRBm×qQ, denote

    ˜M=G3G4(I4mnW[4nq,4n2])(ki=1vRr(Ai)vRc(Bi))J,

    where G3, G4 have the same structure as G in Theorem 3.3 excepting the dimension. Hence the set SHQ of Problem 2 is represented as

    SHQ={X|vRs(X)=˜MvRr(C)+(I2n2n˜M˜M)y,  yR2n2n}. (4.3)

    And then, the minimal norm solution XHQ of Problem 2 satisfies

    vRs(XHQ)=˜MvRr(C). (4.4)

    Proof.

    ki=1AiXBi=CvRr(ki=1AiXBi)=vRr(C).

    Using Theorem 4.4, we can obtain

    vRr(ki=1AiXBi)=vRr(C)G3G4(I4mnW[4nq,4n2])(ki=1vRr(Ai)vRc(Bi))vRc(X)=vRr(C)G3G4(I4mnW[4nq,4n2])(ki=1vRr(Ai)vRc(Bi))JvRs(X)=vRr(C)˜MvRs(X)=vRr(C).

    For the real matrix equation

    ˜MvRs(X)=vRr(C),

    we can obatin vRs(X)=˜MvRr(C)+(I2n2n˜M˜M)y by using Lemma 4.1. Thus we get the formula (4.3).

    Notice

    minXRBn×nHQXminvRs(X)R2n2nvRs(X),

    we obtain that the minimal norm reduced biquaternion Hermitian solution XHQSHQ of Problem 2 satisfies

    vRs(XHQ)=˜MvRr(C).

    Therefore, (4.4) holds.

    Corollary 4.6. Let AiRBm×nQ, BiRBn×qQ, CRBm×qQ, ˜M is given in Theorem 4.5. Then the following statements are equivalent:

    (c) Problem 2 has a solution XSHQ;

    (d) (˜M˜MI4mq)vRr(C)=0.

    Moreover, if (c) holds, the solution set of (1.1) over RBn×nHQ can be represented as

    SHQ={XvRs(X)=˜MvRr(C)+(I2n2n˜MM)y,  yR2n2n}.

    Remark 4.2. When X is a Hermitian matrix, Theorem 4.4 can be used to transform the reduced biquaternion matrix equation ni=1AiXBi=C into real matrix equation ˜MvRs(X)=vRr(C). Corollary 4.6 can be obtained by a proof method similar to Corollary 4.3.

    Similar to Theorem 4.4, we can give the relationship between vRs(X) and vRc(X) in the reduced biquaternion Anti-Hermitian matrix for studying Problem 3.

    Theorem 4.7. Let X=[x11x1nxn1xnn]RBn×nAQ, xij=x1ij+x2iji+x3ijj+x4ijkRBQ, denote

    ˜LXi=[x2iix3iix4iivR(xi(i+1))vR(xin)],  vRs(X)=[˜LX1˜LX2˜LXn],  J=[J1JrJn],  Jr=[J1rJmrJnr],
    Jmrm<r={(δσ+ξ2n2+n)T(δσ+ξ+12n2+n)T(δσ+ξ+22n2+n)T(δσ+ξ+32n2+n)T, Jmrm=r={01×2n2+n(δτ+12n2+n)T(δτ+22n2+n)T(δτ+32n2+n)T, Jmrm>r={(δτξ2n2+n)T(δτξ+12n2+n)T(δτξ+22n2+n)T(δτξ+32n2+n)T,

    where, σ=(m1)(8n+64m)2, τ=(r1)(8n+64r)2, ξ=4(rm).Then

    vRc(X)=JvRs(X).

    Theorem 4.8. Let AiRBm×nQ, BiRBn×qQ, CRBm×qQ, denote

    ˆM=G3G4(I4mnW[4nq,4n2])(ki=1vRr(Ai)vRc(Bi))J,

    where, G3, G4 are defined in Theorem 4.5. Hence the set SAQ of Problem 3 is represented as

    SAQ={X|vRs(X)=ˆMvRr(C)+(I2n2+nˆMˆM)y,  yR2n2+n}. (4.5)

    And the minimal norm reduced biquaternion Anti-Hermitian solution XAQSAQ of Problem 3 satisfies

    vRs(XAQ)=ˆMvRr(C). (4.6)

    Corollary 4.9. Let AiRBm×nQ, BiRBn×qQ, CRBm×qQ, ˆM is given in Theorem 4.8. Then the following statements are equivalent:

    (e) Problem 3 has a solution XSAQ;

    (f) (ˆMˆMI4mq)vRr(C)=0.

    Moreover, if (f) holds, the solution set of (1.1) over RBn×nAQ can be represented as

    SAQ={X|vRs(X)=ˆMvRr(C)+(I2n2+nˆMˆM)y,  yR2n2+n}.

    Based on the algebraic solutions in Section 4, we now present the numerical algorithms and numerical examples for finding solutions of Problems 1–3 in this section.

    To simplify the process of checking the algorithms, in the following numerical examples, we use the reduced biquaternion matrix equation A1XB1+A2XB2=C. To ensure that Problem 1 under testing has a solution, we suppose Ai, Bi, C, X are known reduced biquaternion matrices.

    Example 5.1. Let m=n=p=q=3, and Ai=Ai1+Ai2i+Ai3j+Ai4kRBm×nQ, Bi=Bi1+Bi2i+Bi3j+Bi4kRBp×qQ(i = 1, 2), X=X1+X2i+X3j+X4kRBn×pQ. We take

    A11=[524376728], A12=[153945845], A13=[844889659], A14=[622638652],
    A21=[244232299], A22=[1033446167], A23=[231143358], A24=[0559610725],
    B11=[5642710540], B12=[917931837], B13=[189773597], B14=[256056799],
    B21=[8256922010], B22=[711575501], B23=[8110876758], B24=[514418828]
    X1=[143474560], X2=[10432210159], X3=[1447510393], X4=[7717710522].

    Compute

    C=A1XB1+A2XB2. (5.1)

    Denote ε1=log10MMI4mq, we obtain

    rank(M)=36, ε1=12.6916.

    According to Algorithm 5.1, the reduced biquaternion matrix equation (5.1) has a unique solution XQSQ, we can get ε2=log10XQX=11.3929.

    Algorithm 5.1. For Problem 1
    Step1: Input: Ai, Bi, C, W[4pq,4np] AiRBm×nQ BiRBp×qQ and CRBm×qQ (i=1:k)
    Step2: Compute vRr(Ai), vRc(Bi), vRr(C), G1, G, M
    Step3: if b in Corollary 4.3 holds, then calculate the solution XSQ according to (4.1).
    Step4: if b in Corollary 4.3 and rank(M)=4np hold, then calculate the unique solution XQ according to (4.2).
    Step5: Output: the solution XSQ

     | Show Table
    DownLoad: CSV

    Example 5.2. Ai and Bi (i=1,2) are defined in Example 5.1. Suppose

    ˜X1=[122205.525.57], ˜X2=[022.52032.530], ˜X3=[021.52001.500], ˜X4=[01.501.501010],
    ˜X=˜X1+˜X2i+˜X3j+˜X4k=[122i2j+1.5k2+2.5i1.5j2+2i+2j1.5k05.53i+k22.5i+1.5j5.5+3ik7]RB3×3HQ

    Compute (5.1). we can get ε3=log10˜M˜MI4mq=10.7394 and rank(˜M)=15. The reduced biquaternion matrix equation (5.1) has a unique solution XHQ by using Algorithm (5.2). So we can get ε4=log10XHQ˜X=13.5758.

    Algorithm 5.2. For Problem 2
    Step1: Input: Ai, Bi, C, W[4nq,4n2] AiRBm×nQ BiRBn×qQ and CRBm×qQ (i=1:k)
    Step2: Compute vRr(Ai), vRc(Bi), vRr(C), G3, G4, ˜M
    Step3: if d in Corollary 4.6 holds, then calculate the solution XSHQ according to (4.3).
    Step4: if b in Corollary 4.6 and rank(˜M)=2n2n hold, then calculate the unique solution XHQ according to (4.4).
    Step5: Output: the solution XSHQ

     | Show Table
    DownLoad: CSV

    Example 5.3. Ai and Bi (i=1,2) are defined in Example 5.1. Suppose

    ¯X1=[00.51.50.5021.520], ¯X2=[175.5744.55.54.55], ¯X3=[865687579], ¯X4=[644436.546.52],
    ¯X=¯X1+¯X2i+¯X3j+¯X4k=[i+8j+6k0.5+7i+6j+4k1.5+5.5i+5j+4k0.5+7i+6j+4k4i+8j+3k2+4.5i+7j+6.5k1.5+5.5i+5j+4k2+4.5i+7j+6.5k5i+9j+2k]RB3×3AQ

    Similarly, Compute (5.1). we can get ε5=log10ˆMˆMI4mq=9.6723 and rank(˜M)=21. The reduced biquaternion matrix equation (5.1). has a unique solution XAQ by using Algorithm 5.3. So we can get ε6=log10XAQ¯X=12.6248.

    Algorithm 5.3. For Problem 3
    Step1: Input: Ai, Bi, C, W[4nq,4n2] AiRBm×nQ BiRBn×qQ and CRBm×qQ (i=1:k)
    Step2: Compute vRr(Ai), vRc(Bi), vRr(C), G3, G4, ˆM
    Step3: if f in Corollary 4.9 holds, then calculate the solution XSAQ according to (4.5).
    Step4: if f in Corollary 4.9 and rank(M)=2n2+n hold, then calculate the unique solution XAQ according to (4.6).
    Step5: Output: the solution XSAQ

     | Show Table
    DownLoad: CSV

    Example 5.4. For m=n=p=q=2K, Ai and Bi generated randomly for K=2:6. Consider the reduced biquaternion matrix equation (1.1), we record the errors in the three Problems in Figure 1.

    Figure 1.  Errors in different dimensions.

    Examples 5.1–5.4 are used to show the feasibility of Algorithms 5.1–5.3.

    In this paper, we proposed a real vector representation of reduced biquaternion matrix, which preserves the relative positions of the elements in the original reduced biquaternion matrix. For every element in reduced biquaternion, the real and three imaginary parts are remained as a whole. Combined this real vector representation with semi-tensor product of matrices, we solved the Problems 1–3.

    This work was supported by National Natural Science Foundation of China 62176112; Science and Technology Project of Department of Education, Shandong Province ZR2020MA053; the science foundation of Liaocheng University under grants 318011921.

    The authors declare that they have no competing interests.


    Acknowledgments



    The project was partly funded by the Federal Ministry for Economic Affairs and Energy in the scope of the ZIM project ZF4036107 and by the HiF fund of Bielefeld University of Applied Sciences. The APC is funded by the Open Access Publication Fund of Bielefeld University of Applied Sciences. We thank Martina Holt and Karl-Josef Dietz for constructive criticism of the manuscript, lending laboratory equipment and for advice on the implementation of the experiments. Furthermore, we thank Anja Storck for her kind support.

    Conflict of interest



    The authors declare no conflict of interest.

    Author contributions



    Bennet Brockhagen: conceptualization, methodology, formal analysis, investigation, visualization; Fabian Schoden: conceptualization, methodology, formal analysis, investigation, visualization, writing–first draft; Jan Lukas Storck: conceptualization, methodology, formal analysis, investigation, visualization; Timo Grothe: conceptualization, investigation, visualization; Christian Eßelmann: investigation, validation; Robin Böttjer: methodology; Anke Rattenholl: validation; Frank Gudermann: validation. All authors read and substantially modified the manuscript.

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