Research article

Characterizations and properties of hyper-dual Moore-Penrose generalized inverse

  • Received: 10 October 2024 Revised: 30 November 2024 Accepted: 09 December 2024 Published: 16 December 2024
  • MSC : 15A09, 15A24, 15B33

  • In this paper, the definition of the hyper-dual Moore-Penrose generalized inverse of a hyper-dual matrix is introduced. Characterizations for the existence of the hyper-dual Moore-Penrose generalized inverse are given, and a formula for the hyper-dual Moore-Penrose generalized inverse is presented whenever it exists. Least-squares properties of the hyper-dual Moore-Penrose generalized inverse are discussed by introducing a total order of hyper-dual numbers. We also introduce the definition of a dual matrix of order n. A necessary and sufficient condition for the existence of the Moore-Penrose generalized inverse of a dual matrix of order n is given.

    Citation: Qi Xiao, Jin Zhong. Characterizations and properties of hyper-dual Moore-Penrose generalized inverse[J]. AIMS Mathematics, 2024, 9(12): 35125-35150. doi: 10.3934/math.20241670

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  • In this paper, the definition of the hyper-dual Moore-Penrose generalized inverse of a hyper-dual matrix is introduced. Characterizations for the existence of the hyper-dual Moore-Penrose generalized inverse are given, and a formula for the hyper-dual Moore-Penrose generalized inverse is presented whenever it exists. Least-squares properties of the hyper-dual Moore-Penrose generalized inverse are discussed by introducing a total order of hyper-dual numbers. We also introduce the definition of a dual matrix of order n. A necessary and sufficient condition for the existence of the Moore-Penrose generalized inverse of a dual matrix of order n is given.



    Dual numbers were introduced by Clifford [1] in order to expand quaternions to bi-quaternions that represent both rotations and translations. Dual numbers since then have been important and convenient mathematical tools in dealing with some problems in various fields of science and engineering, such as kinematic synthesis [2,3], robotics [4], scara kinematics [5] and displacement analysis [6,7]. A matrix with dual number entries is called a dual matrix. Dual matrices are used today in a variety of fields like kinematic analysis and synthesis of spatial mechanisms, and also in robotics [8]. There are many investigations where the kinematic analysis and synthesis problems are addressed through the solution of overdetermined systems of linear dual equations, and dual generalized inverses of dual matrices have been shown to be very useful in studying the solutions of systems of linear dual equations [9]. For example, the dual Moore-Penrose generalized inverse (DMPGI, for short) provides minimum-norm least-squares solution for the system of linear dual equations[10]

    ˆAˆx=ˆb.

    However, many research results have shown that various dual generalized inverses of dual matrices may not exist. Based on this fact, in the past few years, numerous articles were dedicated to characterizing the existence of different kinds of dual generalized inverses, for example, DMPGI [11,12], weak dual generalized inverse [13], dual core generalized inverse [14,15]. Especially, Wang [16] gave some necessary and sufficient conditions for a dual matrix to have the DMPGI, and a compact formula for the computation of the DMPGI was also given. Zhong and Zhang [17,18] presented some necessary and sufficient conditions for a square dual matrix to have the dual group inverse and the dual Drazin inverse.

    Throughout this paper, we use ˆR to denote the set of dual numbers over the real field. A dual number ˆaˆR has the form

    ˆa=a+ϵa0,

    where a and a0 are real numbers, and ϵ is the dual unity that satisfies the rules

    ϵ0andϵ2=0.

    Hyper-dual numbers are an extension of dual numbers and were first introduced by Fike et al. [19,20,21] to derive the kinematics of a multi-body system. They introduced the hyper-dual numbers to perform second-order numerical differentiation that leads to smaller numerical (subtractive and cancellation) errors as well as to reduced computational time. A hyper-dual number ˜a is a number consisting of four real numbers a0a3 and two dual units ϵ1, ϵ2 with the following rules:

    ϵ21=ϵ22=(ϵ1ϵ2)2=0, ϵ1,ϵ2,ϵ1ϵ20,

    and ˜a is of the form

    ˜a=a0+ϵ1a1+ϵ2a2+ϵ1ϵ2a3. (1.1)

    Notice that we can rewrite the hyper-dual number ˜a in (1.1) as

    ˜a=(a0+ϵ1a1)+ϵ2(a2+ϵ1a3)ˆa+ϵ2ˆa0, (1.2)

    i.e., a hyper-dual number is a combination of two dual numbers, where ˆa is called the primal part and ˆa0 is called the hyper-dual part of ˜a, respectively. In other words, a hyper-dual number can be obtained by replacing the two real numbers in a dual number by two dual numbers. The physical meaning of these two dual numbers in the context of kinematics was discussed in [22,23] by introducing the hyper-dual angle. We denote the set of all hyper-dual numbers over the real field by ˜R. For the sake of convenience, we replace ϵ1, ϵ2 by ϵ, ϵ in (1.1) and (1.2).

    For ˜a˜R, the Taylor series expansion of a dual function of order 2 is given by (see [21])

    f(˜a)=f(a0)+ϵa1f(a0)+ϵa2f(a0)+ϵϵ[a3f(a0)+a1a2f(a0)].

    For example, for a hyper-dual number

    ˜a=a0+ϵa1+ϵa2+ϵϵa3

    with a0>0, the square root of ˜a is given by

    ˜a=a0+ϵa12a0+ϵ[a22a0+ϵ(a32a0a1a24a30)]. (1.3)

    According to (1.3), for

    ˜a=a0+ϵa1+ϵa2+ϵϵa3˜R

    with a00, the absolute value and the Euclidean norm of ˜a can be respectively defined by

    |˜a|=|a0|+ϵsgn(a0)a1+ϵsgn(a0)a2+ϵϵsgn(a0)a3

    and

    ˜a=a0+ϵaT0a1a0+ϵaT0a2a0+ϵϵ(aT0a3+aT1a2a0aT0a1aT0a2a03).

    A matrix with hyper-dual number entries is called a hyper-dual matrix. Analogous to the forms of hyper-dual numbers, an m×n hyper-dual matrix ˜A is defined as

    ˜A=A0+ϵA1+ϵA2+ϵϵA3=(A0+ϵA1)+ϵ(A2+ϵA3)ˆA+ϵˆA0,

    where A0A3 are m×n real matrices, and ϵ and ϵ are dual units. The set of all m×n hyper-dual matrices over the real field is denoted by ˜Rm×n. Some studies on hyper-dual matrices can be found in [24,25].

    For a given hyper-dual matrix ˜A˜Rm×n, if there exists a hyper-dual matrix ˜X˜Rn×m satisfying

    ˜A˜X˜A=˜A,˜X˜A˜X=˜X,(˜A˜X)T=˜A˜X,(˜X˜A)T=˜X˜A, (1.4)

    then we call ˜X the hyper-dual Moore-Penrose generalized inverse (HDMPGI) of ˜A, and denoted by ˜A.

    In this paper, we aim to give some theoretical findings of HDMPGI. The rest of this paper is organized as follows. In Section 2, we give some necessary and sufficient conditions for a hyper-dual matrix to have the HDMPGI, and present a compact formula for HDMPGI whenever it exists. In Section 3, analogous to the applications of the dual Moore-Penrose generalized inverse in linear dual equations, we discuss the least-squares properties of HDMPGI. In Section 4, based on the forms of dual matrices and hyper-dual matrices, we introduce the definition of dual matrix of order n. We also study the existence of the Moore-Penrose generalized inverse of such matrices. The theoretical results are illustrated by some numerical examples.

    Throughout this paper, we use Rn, ˆRn, and ˜Rn to denote the set of all n-dimensional real column vectors, dual column vectors, and hyper-dual column vectors, respectively. Rm×n, ˆRm×n, and ˜Rm×n are, respectively, the set of all m×n real matrices, dual matrices, and hyper-dual matrices. For a real matrix A, r(A) is the rank of A, the superscript "T" is the transpose of a matrix, and In is the identity of order n. is the Euclidean norm of a vector. We will use

    G

    to mean that we define G to be something.

    The following lemma is well-known as singular value decomposition, which will be a basic tool for proving Theorem 2.1.

    Lemma 1.1. [26] Let ARm×n be such that

    r(A)=r.

    Then, there exist real orthogonal matrices URm×m and VRn×n such that

    A=U[Σ000]VT,

    where ΣRr×r is a diagonal positive definite matrix. Then,

    A=V[Σ1000]UT.

    The following lemma will also be used in the proof of Theorem 2.1, which is a rank equality that involves a special 2×2 block matrix and Moore-Penrose generalized inverse.

    Lemma 1.2. [27] Let ARm×n, BRm×k, and CRl×n. Then,

    r[ABC0]=r(B)+r(C)+r[(ImBB)A(InCC)].

    In this section, we study the existence and computation of the HDMPGI. We first give a necessary and sufficient condition for a hyper-dual matrix to be the HDMPGI of a given hyper-dual matrix, which can be obtained directly from the definition of the HDMPGI in (1.4), and we omit the proof.

    Lemma 2.1. Let

    ˜A=ˆA+ϵˆA0˜Rm×n.

    Then, a hyper-dual matrix

    ˜X=ˆX+ϵˆX0˜Rn×m

    is the HDMPGI of ˜A if and only if

    ˆX=ˆA

    and

    {ˆAˆXˆA0+ˆAˆX0ˆA+ˆA0ˆXˆA=ˆA0,ˆXˆAˆX0+ˆXˆA0ˆX+ˆX0ˆAˆX=ˆX0,(ˆAˆX0+ˆA0ˆX)T=ˆAˆX0+ˆA0ˆX,(ˆXˆA0+ˆX0ˆA)T=ˆXˆA0+ˆX0ˆA.

    Analogous to the DMPGI of dual matrices, the HDMPGI of hyper-dual matrices may not exist. We present some necessary and sufficient conditions for the existence of the HDMPGI in the following theorem. A compact formula for the computation of the HDMPGI is also given whenever it exists.

    Theorem 2.1. Let

    ˜A=ˆA+ϵˆA0=A0+ϵA1+ϵA2+ϵϵA3˜Rm×n.

    Then, the following statements are equivalent:

    (ⅰ) The HDMPGI of ˜A exists;

    (ⅱ)

    ˜A=U[Σ000]VT+ϵU[R1R2R30]VT+ϵ(U[Y1Y2Y30]VT+ϵU[Z1Z2Z3Z4]VT),

    where U and V are real orthogonal matrices of orders m and n, respectively, Σ is a diagonal positive definite matrix, and R1R3, Y1Y3, Z1Z4 are real matrices of appropriate sizes that satisfy

    Z4=R3Σ1Y2+Y3Σ1R2;

    (ⅲ) ˆA exists, and

    (ImˆAˆA)ˆA0(InˆAˆA)=0;

    (ⅳ)

    (ImA0A0)A1(InA0A0)=(ImA0A0)A2(InA0A0)=(ImA0A0)(A3A2A0A1A1A0A2)(InA0A0)=0;

    (ⅴ)

    r[A1A0A00]=r[A2A0A00]=r[A3A2A0A1A1A0A2A0A00]=2r(A0).

    Furthermore, if the HDMPGI of ˜A exists, then

    ˜A=ˆA+ϵ[ˆAˆA0ˆA+(ˆATˆA)ˆA0T(ImˆAˆA)+(InˆAˆA)ˆA0T(ˆAˆAT)]. (2.1)

    Proof. In order to show the equivalence of the five items, we will prove that (ⅰ)(ⅱ), (ⅱ)(ⅲ), (ⅲ)(ⅳ), and (ⅳ)(ⅴ).

    (ⅰ)(ⅱ): If the HDMPGI of

    ˜A=ˆA+ϵˆA0

    exists, then we may assume that

    ˜A=ˆX+ϵˆX0.

    It follows from Lemma 2.1 that the DMPGI of ˆA exists and

    ˆX=ˆA.

    Then, by [16], using the singular value decomposition of real matrices in Lemma 1.1, ˆA and ˆX have the forms

    ˆA=U[Σ000]VT+ϵU[R1R2R30]VT (2.2)

    and

    ˆX=V[Σ1000]UT+ϵV[Σ1R1Σ1Σ2RT3RT2Σ20]UT, (2.3)

    where URm×m and VRn×n are real orthogonal matrices, ΣRr×r is a diagonal positive definite matrix,

    r=r(A0),

    and R1R3 are real matrices of appropriate sizes.

    Let

    ˆA0=U[Y1Y2Y3Y4]VT+ϵU[Z1Z2Z3Z4]VT,ˆX0=V[X1X2X3X4]UT+ϵV[W1W2W3W4]UT.

    Then, a direct calculation shows that

    ˆAˆXˆA0=U[Y1Y200]VT+ϵU[Z1+Σ1RT3Y3Z2+Σ1RT3Y4R3Σ1Y1R3Σ1Y2]VT,ˆAˆX0ˆA=U[ΣX1Σ000]VT+ϵU[ΘΣX1R2R3X1Σ0]VT,

    where

    Θ=ΣX1R1+ΣX2R3+ΣW1Σ+R1X1Σ+R2X3Σ.
    ˆA0ˆXˆA=U[Y10Y30]VT+ϵU[Z1+Y2RT2Σ1Y1Σ1R2Z3+Y4RT2Σ1Y3Σ1R2]VT.

    Hence,

    ˆAˆXˆA0+ˆAˆX0ˆA+ˆA0ˆXˆA=U[2Y1+ΣX1ΣY2Y30]VT+ϵU[Γ1Γ2Γ3Γ4]VT,

    where

    {Γ1=2Z1+Σ1RT3Y3+Y2RT2Σ1+Θ,Γ2=Z2+Σ1RT3Y4+ΣX1R2+Y1Σ1R2,Γ3=Z3+Y4RT2Σ1+R3Σ1Y1+R3X1Σ,Γ4=R3Σ1Y2+Y3Σ1R2.

    According to Lemma 2.1,

    ˆAˆXˆA0+ˆAˆX0ˆA+ˆA0ˆXˆA=ˆA0,

    i.e.,

    U[2Y1+ΣX1ΣY2Y30]VT+ϵU[Γ1Γ2Γ3Γ4]VT=U[Y1Y2Y3Y4]VT+ϵU[Z1Z2Z3Z4]VT.

    Equating the real part and the dual part of both sides of the above equality yields

    Y4=0

    and

    Γ4=R3Σ1Y2+Y3Σ1R2=Z4.

    Therefore, ˜A has the form

    ˜A=U[Σ000]VT+ϵU[R1R2R30]VT+ϵ(U[Y1Y2Y30]VT+ϵU[Z1Z2Z3Z4]VT).

    Conversely, if

    ˜A=U[Σ000]VT+ϵU[R1R2R30]VT+ϵ(U[Y1Y2Y30]VT+ϵU[Z1Z2Z3Z4]VT),

    where U and V are real orthogonal matrices of orders m and n, respectively, Σ is a diagonal positive definite matrix, and

    Z4=R3Σ1Y2+Y3Σ1R2.

    Let

    ˜G=V[Σ1000]UT+ϵV[Σ1R1Σ1Σ2RT3RT2Σ20]UT+ϵ(V[Σ1Y1Σ1Σ2YT3YT2Σ20]UT+ϵV[M1M2M3M4]UT), (2.4)

    where

    {M1=Σ2RT3Y3Σ1Σ1Y2RT2Σ2Σ2YT3R3Σ1Σ1R2YT2Σ2,M2=Σ2ZT3Σ2RT1Σ1YT3Σ1R1Σ2YT3Σ2YT1Σ1RT3Σ1Y1Σ2RT3,M3=ZT2Σ2RT2Σ1YT1Σ2YT2Σ2R1Σ1YT2Σ1RT1Σ2RT2Σ2Y1Σ1,M4=RT2Σ3YT3+YT2Σ3RT3.

    Then,

    ˜A˜G=U[Ir000]UT+ϵU[0Σ1RT3R3Σ10]UT+ϵU[0Σ1YT3Y3Σ10]UT+ϵϵU[N1N2N3N4]UT,

    where

    {N1=Σ1RT3Y3Σ1Σ1YT3R3Σ1,N2=Σ1ZT3Σ1RT1Σ1YT3Σ1YT1Σ1RT3,N3=Z3Σ1Y3Σ1R1Σ1R3Σ1Y1Σ1,N4=Y3Σ2RT3+R3Σ2YT3.
    ˜G˜A=V[Ir000]VT+ϵV[0Σ1R2RT2Σ10]VT+ϵV[0Σ1Y2YT2Σ10]VT+ϵϵV[P1P2P3P4]VT,

    where

    {P1=Σ1Y2RT2Σ1Σ1R2YT2Σ1,P2=Σ1Z2Σ1Y1Σ1R2Σ1R1Σ1Y2,P3=ZT2Σ1RT2Σ1YT1Σ1YT2Σ1RT1Σ1,P4=YT2Σ2R2+RT2Σ2Y2.

    Then, ˜A˜G and ˜G˜A are symmetric.

    Furthermore,

    ˜A˜G˜A=U[Σ000]VT+ϵU[00R30]VT+ϵU[00Y30]VT+ϵϵU[Σ1RT3Y3Σ1YT3R30Z3Y3Σ1R1R3Σ1Y10]VT+ϵU[R1R200]VT+ϵϵU[Σ1YT3R30Y3Σ1R1Y3Σ1R2]VT+ϵϵU[Σ1RT3Y30R3Σ1Y1R3Σ1Y2]VT+ϵU[Y1Y200]VT+ϵϵU[Z1Z200]VT=U[Σ000]VT+ϵU[R1R2R30]VT+ϵU[Y1Y2Y30]VT+ϵϵU[Z1Z2Z3Z4]VT=˜A,˜G˜A˜G=V[Σ1000]UT+ϵV[00RT2Σ20]UT+ϵV[00YT2Σ20]UT+ϵϵV[Σ1Y2RT2Σ2Σ1R2YT2Σ20ZT2Σ2RT2Σ1YT1Σ2YT2Σ1RT1Σ20]UT+ϵV[Σ1R1Σ1Σ2RT300]UT+ϵϵV[Σ1Y2RT2Σ20YT2Σ1RT1Σ1YT2Σ3RT3]UT+ϵV[Σ1Y1Σ1Σ2YT300]UT+ϵϵV[Σ1R2YT2Σ20RT2Σ1YT1Σ1RT2Σ3YT3]UT+ϵϵV[M1M200]UT=V[Σ1000]UT+ϵV[Σ1R1Σ1Σ2RT3RT2Σ20]UT+ϵV[Σ1Y1Σ1Σ2YT3YT2Σ20]UT+ϵϵV[M1M2M3M4]UT=˜G.

    Hence, ˜A and ˜G satisfy the four Penrose conditions in (1.4), i.e., ˜G is the HDMPGI of ˜A.

    (ⅱ)(ⅲ): If

    ˜A=ˆA+ϵˆA0,

    where

    ˆA=U[Σ000]VT+ϵU[R1R2R30]VT,     ˆA0=U[Y1Y2Y30]VT+ϵU[Z1Z2Z3Z4]VT,

    then by [16], the DMPGI of ˆA exists and ˆA has the matrix form in (2.3). Substituting the matrix forms of ˆA, ˆA, and ˆA0 into (ImˆAˆA)ˆA0(InˆAˆA), we obtain

    (ImˆAˆA)ˆA0(InˆAˆA)=(U[000Imr]UTϵU[0Σ1RT3R3Σ10]UT)×(U[Y1Y2Y30]VT+ϵU[Z1Z2Z3Z4]VT)×(V[000Inr]VTϵV[0Σ1R2RT2Σ10]VT)=ϵU[000Z4R3Σ1Y2Y3Σ1R2]VT.

    Therefore, if

    Z4=R3Σ1Y2+Y3Σ1R2,

    then

    (ImˆAˆA)ˆA0(InˆAˆA)=0.

    On the other hand, if ˆA exists, then ˆA and ˆA have the matrix forms in (2.2) and (2.3), respectively. By a direct calculation, we have

    (ImˆAˆA)ˆA0(InˆAˆA)=U[000Y4]VT+ϵU[0Σ1RT3Y4Y4RT2Σ1Z4R3Σ1Y2Y3Σ1R2]VT.

    Hence, if

    (ImˆAˆA)ˆA0(InˆAˆA)=0,

    then Y4=0 and

    Z4=R3Σ1Y2+Y3Σ1R2.

    (ⅲ)(ⅳ): By [16], if ˆA exists, then

    (ImA0A0)A1(InA0A0)=0.

    Moreover, if

    (ImˆAˆA)ˆA0(InˆAˆA)=0,

    then substituting

    ˆA=A0+ϵA1,    ˆA0=A2+ϵA3

    and

    ˆA=A0+ϵ[A0A1A0+(AT0A0)AT1(ImA0A0)+(InA0A0)AT1(A0AT0)]

    into

    (ImˆAˆA)ˆA0(InˆAˆA)=0

    gives

    (ImA0A0)A2(InA0A0)+ϵ[(ImA0A0)(A3A2A0A1A1A0A2)(InA0A0)(ImA0A0)A2(InA0A0)AT1(A0AT0)A0A0(AT0A0)AT1(ImA0A0)A2(InA0A0)]=0,

    which implies

    (ImA0A0)A2(InA0A0)=0

    and

    (ImA0A0)(A3A2A0A1A1A0A2)(InA0A0)=0.

    Conversely, if

    (ImA0A0)A1(InA0A0)=0,

    then by [16], ˆA exists. Moreover, if

    (ImA0A0)A2(InA0A0)=0

    and

    (ImA0A0)(A3A2A0A1A1A0A2)(InA0A0)=0,

    then it is not difficult to see that

    (ImˆAˆA)ˆA0(InˆAˆA)=0.

    (ⅳ)(ⅴ): It follows directly from Lemma 1.2.

    It remains to show that

    ˜G=ˆA+ϵ[ˆAˆA0ˆA+(ˆATˆA)ˆA0T(ImˆAˆA)+(InˆAˆA)ˆA0T(ˆAˆAT)].

    By a direct calculation, we have

    ˆA=V[Σ1000]UT+ϵV[Σ1R1Σ1Σ2RT3RT2Σ20]UT, (2.5)
    ˆAˆA0ˆA=V[Σ1Y1Σ1000]UT+ϵV[Q1Σ1Y1Σ2RT3RT2Σ2Y1Σ10]UT, (2.6)

    where

    Q1=Σ1(R1Σ1Y1Σ1Y1Σ1R1Σ1+Z1Σ1+Σ1RT3Y3Σ1+Y2RT2Σ2).
    (ˆATˆA)ˆA0T(ImˆAˆA)=V[0Σ2YT300]UT+ϵV[Σ2YT3R3Σ1Q20RT2Σ3YT3]UT,

    where

    Q2=Σ2ZT3Σ2RT1Σ1YT3Σ1R1Σ2YT3Σ2YT1Σ1RT3.
    (InˆAˆA)ˆA0T(ˆAˆAT)=V[00YT2Σ20]UT+ϵV[Σ1R2YT2Σ20Q3YT2Σ3RT3]UT, (2.7)

    where

    Q3=ZT2Σ2RT2Σ1YT1Σ2YT2Σ2R1Σ1YT2Σ1RT1Σ2.

    Now, it can be seen from (2.4)–(2.7) that

    ˜G=ˆA+ϵ[ˆAˆA0ˆA+(ˆATˆA)ˆA0T(ImˆAˆA)+(InˆAˆA)ˆA0T(ˆAˆAT)],

    and thus ˜A has the expression in (2.1).

    Remark that we can know whether the HDMPGI of a hyper-dual matrix exists by checking one of the four conditions in Theorem 2.1, especially by condition (ⅴ). Once the HDMPGI exists, we can obtain it by the formula given in (2.1). We illustrate this by the following example:

    Example 2.1. Let

    ˜A=[1100]+ϵ[1211]+ϵ[0022]+ϵϵ[1113]A0+ϵA1+ϵA2+ϵϵA3.

    Since

    r[A1A0A00]=r[A2A0A00]=r[A3A2A0A1A1A0A2A0A00]=2=2r(A0),

    then by Theorem 2.1(ⅴ), the HDMPGI of ˜A exists.

    A direct computation shows that

    ˆA=(A0+ϵA1)=[120120]+ϵ[1121212]

    and

    ˜A=ˆA+ϵ[ˆAˆA0ˆA+(ˆATˆA)ˆA0T(ImˆAˆA)+(InˆAˆA)ˆA0T(ˆAˆAT)]=[120120]+ϵ[1121212]+ϵ[0101]+ϵϵ[52523232].

    Qi et al. [28] introduced a total order over ˆR. Suppose

    ˆp=p+ϵp0,    ˆq=q+ϵq0ˆR.

    We have ˆp<ˆq if p<q, or p=q and p0<q0; ˆp=ˆq if p=q and p0=q0. The total order provides an efficient way to compare the magnitude of two dual numbers. Based on the total order over ˆR, Wang et al. [29] extended it to dual vectors and introduced a QLY total order Q over ˆRm. We introduce a total order over ˜R as follows. For two hyper-dual numbers

    ˜p=ˆp+ϵˆp0,    ˜q=ˆq+ϵˆq0˜R.

    We have ˜p<˜q if ˆp<ˆq, or ˆp=ˆq and ˆp0<ˆq0; ˜p=˜q if ˆp=ˆq and ˆp0=ˆq0. If ˜a>0, then we say that ˜a is a positive hyper-dual number. If ˜a0, then we call ˜a a nonnegative hyper-dual number.

    Recall that for a dual vector

    ˆx=x+ϵx0ˆRn,

    the Euclidean norm of ˆx is defined as [28]

    ˆx={x+2ϵxTx0x,ifx0,x0ϵ,ifx=0.

    For a hyper-dual number ˜a, ˜a2 is also a hyper-dual number. We may study least-squares properties of HDMPGI by the total order. However, ˜a2 is not always nonnegative, for example,

    ϵa1+ϵa2+ϵϵa32=(ϵa1+ϵa2+ϵϵa3)T(ϵa1+ϵa2+ϵϵa3)=2ϵϵaT1a2.

    For this reason, we introduce the following set:

    ˜R0m={a0+ϵa1+ϵa2+ϵϵa3a0,a1,a2,a3Rm,a00ora0=0andaT1a20}.

    For a hyper-dual vector

    ˜a=a0+ϵa1+ϵa2+ϵϵa3˜Rm,˜a2=˜aT˜a=a02+2ϵaT0a1+2ϵaT0a2+2ϵϵ(aT0a3+aT1a2).

    Hence, if ˜a˜R0m, then ˜a20.

    For ˜a˜R0m, we define the Euclidean norm of ˜a as follows:

    ˜a={a0+ϵaT0a1a0+ϵaT0a2a0+ϵϵ(aT0a3+aT1a2a0aT0a1aT0a2a03),ifa00,ϵaT1a2+ϵaT1a2+ϵϵa3,ifa0=0,a10,a20andaT1a20,ϵa1+2ϵϵaT1a3a1,ifa0=a2=0,a10,ϵa2+2ϵϵaT2a3a2,ifa0=a1=0,a20,ϵϵa3,ifa0=a1=a2=0,0,ifa0=a1=a2=a3=0. (3.1)

    Upon expansion into its primal and hyper-dual parts, the system of linear hyper-dual equations

    ˜A˜x=˜b

    reveals four systems of real linear equations,

    {A0x0=b0,A0x1=b1A1x0,A0x2=b2A2x0,A0x3=b3A3x0A2x1A1x2. (3.2)

    We will consider the least-squares solutions of the system of linear hyper-dual equations

    ˜A˜x=˜b

    under some constraints. We suppose that the real linear equation

    A0x0=b0

    in (3.2) is inconsistent, and thus

    ˜A˜x=˜b

    is also inconsistent. Remark that the symbol ˜A(1,3) is the set of hyper-dual matrices ˜X that satisfies the two equations

    ˜A˜X˜A=˜A

    and

    (˜A˜X)T=˜A˜X

    in (1.4), which is important for studying least-squares solutions of systems of linear hyper-dual equations.

    Theorem 3.1. Let ˜A˜Rm×n be such that ˜A exists, ˜b˜Rm, and

    (˜A˜A(1,3)Im)˜b˜R0m.

    Denote

    ˜x0=˜A(1,3)˜b(In˜A(1,3)˜A)˜w˜Rn,

    where ˜w˜Rn is an arbitrary hyper-dual vector. Then,

    ˜A˜x0˜b˜A˜x˜b

    for any hyper-dual vector ˜x that satisfies

    ˜A(˜x˜A(1,3)˜b)˜R0m.

    Proof. Adding and subtracting ˜A˜A(1,3)˜b, we get

    ˜e=˜A˜x˜b=˜A(˜x˜A(1,3)˜b)+(˜A˜A(1,3)˜b˜b)˜u+˜v. (3.3)

    Since

    ˜vT˜u=˜bT(˜A˜A(1,3)Im)˜A(˜x˜A(1,3)˜b)=0

    in (3.3), then ˜uT˜v is also zero and

    ˜e2=˜u+˜v2=(˜u+˜v)T(˜u+˜v)=˜u2+˜v2+2˜uT˜v=˜u2+˜v2. (3.4)

    Let

    ˜u=u0+ϵu1+ϵu2+ϵϵu3.

    Then,

    ˜u2=˜uT˜u=u02+2ϵuT0u1+2ϵuT0u2+2ϵϵ(uT0u3+uT1u2). (3.5)

    If ˜u˜R0m, then it can be observed from (3.5) that

    ˜u20,

    and thus

    ˜e2˜v2

    by (3.4), and equality holds if and only if

    ˜u2=0.

    Let

    ˜e=e0+ϵe1+ϵe2+ϵϵe3,   ˜v=v0+ϵv1+ϵv2+ϵϵv3.

    Then,

    ˜e2=e02+2ϵeT0e1+2ϵeT0e2+2ϵϵ(eT0e3+eT1e2), (3.6)
    ˜v2=v02+2ϵvT0v1+2ϵvT0v2+2ϵϵ(vT0v3+vT1v2). (3.7)

    Since the system of real linear equations

    A0x0=b0

    is inconsistent, then e00, and thus

    ˜e2>0.

    In this case, it follows from (3.1) that

    ˜e=e0+ϵeT0e1e0+ϵeT0e2e0+ϵϵ(eT0e3+eT1e2e0eT0e1eT0e2e03). (3.8)

    By the assumption, ˜v˜R0m, and then

    ˜v20.

    We consider the following two cases:

    Case 1. ˜v2>0. In this case, either v00 or v0=0 and vT1v2>0. If v0=0 and vT1v2>0, then by (3.1),

    ˜v=ϵvT1v2+ϵvT1v2+ϵϵv3.

    Hence, by (3.8), ˜e>˜v.

    If v00, then

    ˜v=v0+ϵvT0v1v0+ϵvT0v2v0+ϵϵ(vT0v3+vT1v2v0vT0v1vT0v2v03). (3.9)

    Subcase 1. ˜e2>˜v2.

    In this case, by (3.6) and (3.7),

    e0>v0ore0=v0,
    eT0e1>vT0v1ore0=v0,
    eT0e1=vT0v1,eT0e2>vT0v2ore0=v0,
    eT0e1=vT0v1,    eT0e2=vT0v2,    eT0e3+eT1e2>vT0v3+vT1v2.

    Then, it can be observed from (3.8) and (3.9) that ˜e>˜v.

    Subcase 2. ˜e2=˜v2.

    In this case,

    e0=v0,   eT0e1=vT0v1,   eT0e2=vT0v2

    and

    eT0e3+eT1e2=vT0v3+vT1v2.

    Hence, it can be easily seen from (3.8) and (3.9) that ˜e=˜v.

    Case 2. ˜e2>˜v2=0.

    By the assumption, ˜v˜R0m. If ˜v2=0, then by (3.7), v0=0 and

    vT1v2=0.

    We need only to consider the following five subcases:

    (ⅰ) v0=0, v10, v20, vT1v2=0. In this subcase, by (3.1),

    ˜v=ϵϵv3.

    (ⅱ) v0=v1=0, v20. In this subcase, by (3.1),

    ˜v=ϵv2+2ϵϵvT2v3v2.

    (ⅲ) v0=v2=0, v10. In this subcase, by (3.1),

    ˜v=ϵv1+2ϵϵvT1v3v1.

    (ⅳ) v0=v1=v2=0. In this subcase, by (3.1),

    ˜v=ϵϵv3.

    (ⅴ) v0=v1=v2=v3=0. In this subcase, by (3.1), ˜v=0.

    For all these five subcases, by the total order defined above,

    ˜e>˜v.

    Therefore, if

    ˜A˜A(1,3)˜b˜b˜R0m,

    then

    ˜A˜x0˜b=˜A[˜A(1,3)˜b(In˜A(1,3)˜A)˜w]˜b=˜A˜A(1,3)˜b˜b˜A˜x˜b

    for any ˜x that satisfies

    ˜A(˜x˜A(1,3)˜b)˜R0m.

    This completes the proof.

    Theorem 3.1 gives an analogous result to those of the least-squares problem of linear real equations and linear dual equations. It should be noted that the condition

    ˜u20

    is necessary for studying least-squares problem of linear hyper-dual equations, and this is the reason why we introduce the vector set ˜R0m and the total order over ˜R.

    Example 3.1. Consider the inconsistent hyper-dual equation

    ˜A˜x˜b,

    where ˜A is the hyper-dual matrix in Example 2.1, and

    ˜b=[2.87.3]+ϵ[1.65.3]+ϵ[21.618.5]+ϵϵ[31.235.2].

    Then, a direct calculation shows that

    (˜A˜A(1,3)In)˜b=(˜A˜AIn)˜b=[07.3]+ϵ[7.32.5]+ϵ[14.612.9]+ϵϵ[10.616]˜R02.

    Let

    ˜x1=[1.64.3]+ϵ[16.32.8]+ϵ[8.37.6]+ϵϵ[6.222.6].

    Then,

    ˜A(˜x1˜A(1,3)˜b)=˜A˜x1˜A˜A(1,3)˜b=˜A˜x1˜A˜A˜b=[3.10]+ϵ[20.43.1]+ϵ[20.36.2]+ϵϵ[13.217.4]˜R02

    and

    ˜A˜x1˜b=[3.17.3]+ϵ[27.70.6]+ϵ[5.76.7]+ϵϵ[23.833.4].

    Therefore, by (3.1),

    ˜A˜A(1,3)˜b˜b=˜A˜A˜b˜b=7.3+ϵ2.5+ϵ12.9ϵϵ1.4

    and

    ˜A˜x1˜b=7.93+ϵ10.3+ϵ4ϵϵ47.

    Now, by the total order,

    ˜A˜A(1,3)˜b˜b<˜A˜x1˜b.

    We choose another hyper-dual vector ˜x2 as follows:

    ˜x2=[1.61.2]+ϵ[2.52.8]+ϵ[11.66.8]+ϵϵ[24.632.2].

    Then,

    ˜A(˜x2˜A(1,3)˜b)=˜A˜x2˜A˜A(1,3)˜b=˜A˜x2˜A˜A˜b=ϵ[10.20]+ϵ[31.40]+ϵϵ[51.851.8]˜R02

    and

    ˜A˜x2˜b=[07.3]+ϵ[2.92.5]+ϵ[16.812.9]+ϵϵ[41.235.8].

    It follows from (3.1) that

    ˜A˜x2˜b=7.3+ϵ2.5+ϵ12.9+ϵϵ42.5.

    Hence,

    ˜A˜A(1,3)˜b˜b<˜A˜x2˜b.

    Corollary 3.1. Let ˆAˆRm×n be such that ˆA exists, ˆbˆRm. Denote

    ˆx0=ˆA(1,3)ˆb(InˆA(1,3)ˆA)ˆw,

    where ˆwˆRm is an arbitrary dual vector. Then,

    ˆAˆx0ˆbˆAˆxˆb

    for all ˆxˆRn.

    For a hyper-dual number

    ˜a=a0+ϵa1+ϵa2+ϵϵa3,

    if a00, then we say that ˜a is appreciable. Appreciable hyper-dual vectors and appreciable hyper-dual matrices can be defined similarly. We now consider minimum-norm least-squares solution of

    ˜A˜x=˜b

    under some certain restrictions.

    Theorem 3.2. Let ˜A˜Rm×n be such that ˜A exists, ˜b˜Rm, and ˜A˜b˜R0n. If ˜A˜A˜b is appreciable, then

    ˜A˜b˜A˜b+(In˜A˜A)˜h

    for any hyper-dual vector ˜h that satisfies

    (In˜A˜A)˜h˜R0n.

    Proof. Since

    [(In˜A˜A)˜h]T˜A˜b=˜hT(In˜A˜A)˜A˜b=0,

    then

    ˜A˜b+(In˜A˜A)˜h2=˜A˜b2+(In˜A˜A)˜h2. (3.10)

    If a hyper-dual vector ˜h satisfies

    (In˜A˜A)˜h˜R0n,

    then

    (In˜A˜A)˜h20.

    Hence, it can be observed from (3.10) that

    ˜A˜b+(In˜A˜A)˜h2˜A˜b2.

    On the other hand, let

    ˜A=A0+ϵA1+ϵA2+ϵϵA3,     ˜A˜b+(In˜A˜A)˜h=x0+ϵx1+ϵx2+ϵϵx3.

    Then,

    ˜A˜A˜b=˜A[˜A˜b+(In˜A˜A)˜h]=A0x0+ϵ(A0x1+A1x0)ϵ(A0x2+A2x0)+ϵϵ(A0x3+A3x0+A1x2+A2x1). (3.11)

    If ˜A˜A˜b is appreciable, it follows from (3.11) that A0x00. Hence, x00 and ˜A˜b+(In˜A˜A)˜h is appreciable. In this case,

    ˜A˜b+(In˜A˜A)˜h2>0.

    Moreover, ˜A˜b˜R0n implies

    ˜A˜b20.

    Therefore, by an analogous discussion as the proof of Theorem 3.1, we conclude that

    ˜A˜b˜A˜b+(In˜A˜A)˜h.

    This completes the proof.

    Corollary 3.2. Let ˆAˆRm×n be such that ˆA exists, ˆbˆRm. If ˆAˆAˆb is appreciable, then

    ˆAˆbˆAˆb+(InˆAˆA)ˆh

    for all ˆhˆRn.

    Dual matrices and hyper-dual matrices may be referred to as dual matrices of orders 1 and 2, respectively. Specifically, real matrices are of order 0. Then, a dual matrix in ˆRm×n is constituted of two dual matrices of order 0, and a hyper-dual matrix in ˜Rm×n is constituted of two dual matrices of order 1. From this perspective, we define a dual matrix of order n as follows:

    ˆA(n)=ˆB(n1)+ϵnˆC(n1),

    where ˆB(n1) and ˆC(n1) are two dual matrices of order n1, and ϵn is a dual unit. Hence, a dual matrix of order n can be obtained by two dual matrices of order n1. For example, a dual matrix of order 3 is of the form

    ˆA(3)=ˆB(2)+ϵ3ˆC(2)=A0+ϵ1A1+ϵ2A2+ϵ1ϵ2A3+ϵ3(A4+ϵ1A5+ϵ2A6+ϵ1ϵ2A7).

    In this section, we study the conditions for the existence of the Moore-Penrose generalized inverse of dual matrices of order n. Denote the set of all m×n dual matrices of order n by ˆRm×n(n).

    Theorem 4.1. Let

    ˆA(n)=ˆB(n1)+ϵnˆC(n1)ˆRm×n(n).

    Then, ˆA(n) has a Moore-Penrose generalized inverse if and only if (ˆB(n1)) exists and

    [ImˆB(n1)(ˆB(n1))]ˆC(n1)[In(ˆB(n1))ˆB(n1)]=0.

    Moreover, if the Moore-Penrose generalized inverse of ˆA(n) exists, then

    (ˆA(n))=(ˆB(n1))+ϵnˆZ(n1),

    where

    ˆZ(n1)=(ˆB(n1))ˆC(n1)(ˆB(n1))+[(ˆB(n1))TˆB(n1)](ˆC(n1))T×[ImˆB(n1)(ˆB(n1))]+[In(ˆB(n1))ˆB(n1)](ˆC(n1))T[ˆB(n1)(ˆB(n1))T].

    Proof. If ˆA(n) has a Moore-Penrose generalized inverse, we may suppose that

    ˆX(n)=ˆY(n1)+ϵnˆZ(n1)

    is a Moore-Penrose generalized inverse of ˆA(n). Then, ˆA(n) and ˆX(n) satisfy the four Penrose equations, i.e.,

    ˆA(n)ˆX(n)ˆA(n)=ˆA(n),ˆX(n)ˆA(n)ˆX(n)=ˆX(n),    (ˆA(n)ˆX(n))T=ˆA(n)ˆX(n),(ˆX(n)ˆA(n))T=ˆX(n)ˆA(n).

    Substituting

    ˆA(n)=ˆB(n1)+ϵnˆC(n1)

    and

    ˆX(n)=ˆY(n1)+ϵnˆZ(n1)

    into the above four equations yields

    ˆB(n1)ˆY(n1)ˆB(n1)=ˆB(n1),  ˆY(n1)ˆB(n1)ˆY(n1)=ˆY(n1),(ˆB(n1)ˆY(n1))T=ˆB(n1)ˆY(n1),   (ˆY(n1)ˆB(n1))T=ˆY(n1)ˆB(n1).

    Hence, the Moore-Penrose generalized inverse of ˆB(n1) exists and

    ˆY(n1)=(ˆB(n1)).

    On the other hand, equating the dual parts of both sides of the equation

    ˆA(n)ˆX(n)ˆA(n)=ˆA(n)

    gives

    ˆC(n1)=ˆC(n1)(ˆB(n1))ˆB(n1)+ˆB(n1)(ˆB(n1))ˆC(n1)+ˆB(n1)ˆZ(n1)ˆB(n1),

    which is equivalent to

    ˆB(n1)ˆZ(n1)ˆB(n1)=ˆC(n1)ˆC(n1)(ˆB(n1))ˆB(n1)ˆB(n1)(ˆB(n1))ˆC(n1)ˆD(n1).

    Then,

    ˆD(n1)=ˆB(n1)ˆZ(n1)ˆB(n1)=ˆB(n1)(ˆB(n1))ˆB(n1)ˆZ(n1)ˆB(n1)(ˆB(n1))ˆB(n1)=ˆB(n1)(ˆB(n1))ˆD(n1)(ˆB(n1))ˆB(n1)=ˆB(n1)(ˆB(n1))ˆC(n1)(ˆB(n1))ˆB(n1).

    Now we have

    ˆB(n1)(ˆB(n1))ˆC(n1)(ˆB(n1))ˆB(n1)=ˆC(n1)ˆC(n1)(ˆB(n1))ˆB(n1)ˆB(n1)(ˆB(n1))ˆC(n1),

    that is,

    [ImˆB(n1)(ˆB(n1))]ˆC(n1)[In(ˆB(n1))ˆB(n1)]=0.

    Conversely, if (ˆB(n1)) exists and

    [ImˆB(n1)(ˆB(n1))]ˆC(n1)[In(ˆB(n1))ˆB(n1)]=0,

    then we will show that the Moore-Penrose generalized inverse of ˆA(n) exists, and the matrix

    ˆX(n)=(ˆB(n1))+ϵnˆZ(n1)

    is a Moore-Penrose generalized inverse of ˆA(n), where

    ˆZ(n1)=(ˆB(n1))ˆC(n1)(ˆB(n1))+[(ˆB(n1))TˆB(n1)](ˆC(n1))T×[ImˆB(n1)(ˆB(n1))]+[In(ˆB(n1))ˆB(n1)](ˆC(n1))T[ˆB(n1)(ˆB(n1))T].

    Indeed, by checking the four Penrose equations, we have

    ˆA(n)ˆX(n)ˆA(n)=(ˆB(n1)+ϵnˆC(n1))[(ˆB(n1))+ϵnˆZ(n1)](ˆB(n1)+ϵnˆC(n1))=ˆB(n1)+ϵn[ˆB(n1)(ˆB(n1))ˆC(n1)+ˆC(n1)(ˆB(n1))ˆB(n1)ˆB(n1)(ˆB(n1))ˆC(n1)(ˆB(n1))ˆB(n1)].

    Note that the condition

    [ImˆB(n1)(ˆB(n1))]ˆC(n1)[In(ˆB(n1))ˆB(n1)]=0

    is equivalent to

    ˆC(n1)=ˆB(n1)(ˆB(n1))ˆC(n1)+ˆC(n1)(ˆB(n1))ˆB(n1)ˆB(n1)(ˆB(n1))ˆC(n1)(ˆB(n1))ˆB(n1),

    which means that

    ˆA(n)ˆX(n)ˆA(n)=ˆA(n).

    Moreover,

    ˆX(n)ˆA(n)ˆX(n)=(ˆB(n1))+ϵn{(ˆB(n1))ˆC(n1)(ˆB(n1))+(ˆB(n1))ˆB(n1)[(ˆB(n1))TˆB(n1)](ˆC(n1))T[ImˆB(n1)(ˆB(n1))]+[In(ˆB(n1))ˆB(n1)](ˆC(n1))TˆB(n1)[(ˆB(n1))T]ˆB(n1)(ˆB(n1))}.

    Notice that

    (ˆB(n1))ˆB(n1)[(ˆB(n1))TˆB(n1)]=(ˆB(n1))ˆB(n1)(ˆB(n1))[(ˆB(n1))T]=(ˆB(n1))[(ˆB(n1))T]=[(ˆB(n1))TˆB(n1)]

    and

    [ˆB(n1)(ˆB(n1))T]ˆB(n1)(ˆB(n1))=[(ˆB(n1))T](ˆB(n1))ˆB(n1)(ˆB(n1))=[(ˆB(n1))T](ˆB(n1))=[ˆB(n1)(ˆB(n1))T].

    Therefore,

    ˆX(n)ˆA(n)ˆX(n)=(ˆB(n1))+ϵnˆZ(n1)=ˆX(n).

    Furthermore,

    ˆA(n)ˆX(n)=ˆB(n1)(ˆB(n1))+ϵn[ˆB(n1)ˆZ(n1)+ˆC(n1)(ˆB(n1))]=ˆB(n1)(ˆB(n1))+ϵn{[ImˆB(n1)(ˆB(n1))]ˆC(n1)(ˆB(n1))+[ˆC(n1)(ˆB(n1))]T[ImˆB(n1)(ˆB(n1))]}

    and

    ˆX(n)ˆA(n)=(ˆB(n1))ˆB(n1)+ϵn[(ˆB(n1))ˆC(n1)+ˆZ(n1)ˆB(n1)]=(ˆB(n1))ˆB(n1)+ϵn{(ˆB(n1))ˆC(n1)[Im(ˆB(n1))ˆB(n1)]+[Im(ˆB(n1))ˆB(n1)][(ˆB(n1))ˆC(n1)]T}

    are symmetric, which completes the proof.

    We remark that the necessary and sufficient condition in Theorem 4.1 is a generalization of condition (ⅲ) in Theorem 2.1. However, so far we can not give any other necessary and sufficient conditions due to the complex structure of dual matrices of order n.

    Next, we show the uniqueness of the Moore-Penrose generalized inverse of ˆA(n) whenever it exists.

    Theorem 4.2. Let ˆA(n)ˆRm×n(n). If the Moore-Penrose generalized inverse of ˆA(n) exists, then it is unique.

    Proof. According to the proof of Theorem 4.1, if the Moore-Penrose generalized inverse of

    ˆA(n)=ˆB(n1)+ϵnˆC(n1)

    exists, then the Moore-Penrose generalized inverse of ˆB(n1) exists, and the Moore-Penrose generalized inverse of ˆA(n) is of the form (ˆB(n1))+ϵnˆZ(n1).

    Let

    ˆX(n)1=(ˆB(n1))+ϵnˆZ(n1)1

    and

    ˆX(n)2=(ˆB(n1))+ϵnˆZ(n1)2

    be two Moore-Penrose generalized inverses of ˆA(n). In order to show the uniqueness of the Moore-Penrose generalized inverse of ˆA(n), it suffices to shows that

    ˆZ(n1)1=ˆZ(n1)2.

    Equating the dual part of both sides of the equality

    ˆA(n)ˆX(n)1ˆA(n)=ˆA(n),

    we get

    ˆC(n1)=ˆB(n1)(ˆB(n1))ˆC(n1)+ˆB(n1)ˆZ(n1)1ˆB(n1)+ˆC(n1)(ˆB(n1))ˆB(n1). (4.1)

    Similarly, equating the dual part of both sides of the equality

    ˆA(n)ˆX(n)2ˆA(n)=ˆA(n)

    gives

    ˆC(n1)=ˆB(n1)(ˆB(n1))ˆC(n1)+ˆB(n1)ˆZ(n1)2ˆB(n1)+ˆC(n1)(ˆB(n1))ˆB(n1). (4.2)

    Subtracting (4.1) from (4.2) gives

    ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)ˆB(n1)=0. (4.3)

    On the other hand, equating the dual part of both sides of the equality

    ˆX(n)1ˆA(n)ˆX(n)1=ˆX(n)1

    and the equality

    ˆX(n)2ˆA(n)ˆX(n)2=ˆX(n)2

    respectively yields

    ˆZ(n1)1=(ˆB(n1))ˆB(n1)ˆZ(n1)1+(ˆB(n1))ˆC(n1)(ˆB(n1))+ˆZ(n1)1ˆB(n1)(ˆB(n1)) (4.4)

    and

    ˆZ(n1)2=(ˆB(n1))ˆB(n1)ˆZ(n1)2+(ˆB(n1))ˆC(n1)(ˆB(n1))+ˆZ(n1)2ˆB(n1)(ˆB(n1)). (4.5)

    Then, by subtracting (4.4) from (4.5), we have

    ˆZ(n1)1ˆZ(n1)2=(ˆB(n1))ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)+(ˆZ(n1)1ˆZ(n1)2)ˆB(n1)(ˆB(n1)). (4.6)

    Furthermore, equating the dual part of the equality

    (ˆA(n)ˆX(n)1)T=ˆA(n)ˆX(n)1

    and the equality

    (ˆA(n)ˆX(n)2)T=ˆA(n)ˆX(n)2,

    we have

    [ˆB(n1)ˆZ(n1)1+ˆC(n1)(ˆB(n1))]T=ˆB(n1)ˆZ(n1)1+ˆC(n1)(ˆB(n1))

    and

    [ˆB(n1)ˆZ(n1)2+ˆC(n1)(ˆB(n1))]T=ˆB(n1)ˆZ(n1)2+ˆC(n1)(ˆB(n1)).

    It follows that

    ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)=[ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)]T=(ˆZ(n1)1ˆZ(n1)2)T(ˆB(n1))T=(ˆZ(n1)1ˆZ(n1)2)T(ˆB(n1))T[ˆB(n1)(ˆB(n1))]T=(ˆZ(n1)1ˆZ(n1)2)T(ˆB(n1))TˆB(n1)(ˆB(n1))=[ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)]TˆB(n1)(ˆB(n1))=ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)ˆB(n1)(ˆB(n1)).

    Now, it can be seen from (4.3) that

    ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)=0.

    We can also obtain

    (ˆZ(n1)1ˆZ(n1)2)ˆB(n1)=0

    in a similar way. Substituting

    ˆB(n1)(ˆZ(n1)1ˆZ(n1)2)=0

    and

    (ˆZ(n1)1ˆZ(n1)2)ˆB(n1)=0

    into (4.6), we have

    ˆZ(n1)1=ˆZ(n1)2,

    which completes the proof.

    In this paper, we studied the existence and properties of hyper-dual Moore-Penrose generalized inverse of hyper-dual matrices. We gave several sufficient and necessary conditions for the existence of the HDMPGI of a given hyper-dual matrix. A compact formula for the computation of the HDMPGI was presented whenever it exists. After introducing a total order of hyper-dual numbers and Euclidean norm of a hyper-dual vector in a special set, we studied least-squares solutions and minimum-norm least-squares solutions of systems of linear hyper-dual equations under some certain restrictions. Furthermore, we considered an extension of dual matrices and hyper-dual matrices, i.e., dual matrices of order n. We also gave a sufficient and necessary condition for the existence of the Moore-Penrose generalized inverse of such matrices. The availability of the conditions and formulas obtained in this paper allow the simultaneous solutions of overdetermined systems of linear hyper-dual equations that originate from many kinematic problems. We expect these results will be useful in the future applications. It is also worth considering constructing fast algorithms to find HDMPGI whenever it exists. For example, fast algorithms for finding generalized inverses of complex matrices can be found in [30].

    Qi Xiao: conceptualization, methodology, writing-review and editing, software, validation; Jin Zhong: conceptualization, methodology, writing-original draft, writing-review and editing, validation. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the National Natural Science Foundation of China (Grant No. 12261043), and the Program of Qingjiang Excellent Young Talents, Jiangxi University of Science and Technology (JXUSTQJYX2017007).

    All authors declare no conflicts of interest in this paper.



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