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 ˆa∈ˆR has the form
ˆa=a+ϵa0, |
where a and a0 are real numbers, and ϵ is the dual unity that satisfies the rules
ϵ≠0andϵ2=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 a0–a3 and two dual units ϵ1, ϵ2 with the following rules:
ϵ21=ϵ22=(ϵ1ϵ2)2=0, ϵ1,ϵ2,ϵ1ϵ2≠0, |
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)+ϵ∗a2f′(a0)+ϵϵ∗[a3f′(a0)+a1a2f′′(a0)]. |
For example, for a hyper-dual number
˜a=a0+ϵa1+ϵ∗a2+ϵϵ∗a3 |
with a0>0, the square root of ˜a is given by
√˜a=√a0+ϵa12√a0+ϵ∗[a22√a0+ϵ(a32√a0−a1a24√a30)]. | (1.3) |
According to (1.3), for
˜a=a0+ϵa1+ϵ∗a2+ϵϵ∗a3∈˜R |
with a0≠0, the absolute value and the Euclidean norm of ˜a can be respectively defined by
|˜a|=|a0|+ϵsgn(a0)a1+ϵ∗sgn(a0)a2+ϵϵ∗sgn(a0)a3 |
and
‖˜a‖=‖a0‖+ϵaT0a1‖a0‖+ϵ∗aT0a2‖a0‖+ϵϵ∗(aT0a3+aT1a2‖a0‖−aT0a1aT0a2‖a0‖3). |
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+ϵ∗A2+ϵϵ∗A3=(A0+ϵA1)+ϵ∗(A2+ϵA3)≜ˆA+ϵ∗ˆA0, |
where A0–A3 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 A∈Rm×n be such that
r(A)=r. |
Then, there exist real orthogonal matrices U∈Rm×m and V∈Rn×n such that
A=U[Σ000]VT, |
where Σ∈Rr×r is a diagonal positive definite matrix. Then,
A†=V[Σ−1000]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 A∈Rm×n, B∈Rm×k, and C∈Rl×n. Then,
r[ABC0]=r(B)+r(C)+r[(Im−BB†)A(In−C†C)]. |
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+ϵ∗ˆA0∈˜Rm×n. |
Then, a hyper-dual matrix
˜X=ˆX+ϵ∗ˆX0∈˜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+ϵ∗ˆA0=A0+ϵA1+ϵ∗A2+ϵϵ∗A3∈˜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 R1–R3, Y1–Y3, Z1–Z4 are real matrices of appropriate sizes that satisfy
Z4=R3Σ−1Y2+Y3Σ−1R2; |
(ⅲ) ˆA† exists, and
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=0; |
(ⅳ)
(Im−A0A†0)A1(In−A†0A0)=(Im−A0A†0)A2(In−A†0A0)=(Im−A0A†0)(A3−A2A†0A1−A1A†0A2)(In−A†0A0)=0; |
(ⅴ)
r[A1A0A00]=r[A2A0A00]=r[A3−A2A†0A1−A1A†0A2A0A00]=2r(A0). |
Furthermore, if the HDMPGI of ˜A exists, then
˜A†=ˆA†+ϵ∗[−ˆA†ˆA0ˆA†+(ˆATˆA)†ˆA0T(Im−ˆAˆA†)+(In−ˆA†ˆA)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+ϵ∗ˆA0 |
exists, then we may assume that
˜A†=ˆX+ϵ∗ˆX0. |
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[Σ−1000]UT+ϵV[−Σ−1R1Σ−1Σ−2RT3RT2Σ−20]UT, | (2.3) |
where U∈Rm×m and V∈Rn×n are real orthogonal matrices, Σ∈Rr×r is a diagonal positive definite matrix,
r=r(A0), |
and R1–R3 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+Σ−1RT3Y3Z2+Σ−1RT3Y4R3Σ−1Y1R3Σ−1Y2]VT,ˆAˆX0ˆA=U[ΣX1Σ000]VT+ϵU[ΘΣX1R2R3X1Σ0]VT, |
where
Θ=ΣX1R1+ΣX2R3+ΣW1Σ+R1X1Σ+R2X3Σ. |
ˆA0ˆXˆA=U[Y10Y30]VT+ϵU[Z1+Y2RT2Σ−1Y1Σ−1R2Z3+Y4RT2Σ−1Y3Σ−1R2]VT. |
Hence,
ˆAˆXˆA0+ˆAˆX0ˆA+ˆA0ˆXˆA=U[2Y1+ΣX1ΣY2Y30]VT+ϵU[Γ1Γ2Γ3Γ4]VT, |
where
{Γ1=2Z1+Σ−1RT3Y3+Y2RT2Σ−1+Θ,Γ2=Z2+Σ−1RT3Y4+ΣX1R2+Y1Σ−1R2,Γ3=Z3+Y4RT2Σ−1+R3Σ−1Y1+R3X1Σ,Γ4=R3Σ−1Y2+Y3Σ−1R2. |
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=R3Σ−1Y2+Y3Σ−1R2=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=R3Σ−1Y2+Y3Σ−1R2. |
Let
˜G=V[Σ−1000]UT+ϵV[−Σ−1R1Σ−1Σ−2RT3RT2Σ−20]UT+ϵ∗(V[−Σ−1Y1Σ−1Σ−2YT3YT2Σ−20]UT+ϵV[M1M2M3M4]UT), | (2.4) |
where
{M1=−Σ−2RT3Y3Σ−1−Σ−1Y2RT2Σ−2−Σ−2YT3R3Σ−1−Σ−1R2YT2Σ−2,M2=Σ−2ZT3−Σ−2RT1Σ−1YT3−Σ−1R1Σ−2YT3−Σ−2YT1Σ−1RT3−Σ−1Y1Σ−2RT3,M3=ZT2Σ−2−RT2Σ−1YT1Σ−2−YT2Σ−2R1Σ−1−YT2Σ−1RT1Σ−2−RT2Σ−2Y1Σ−1,M4=RT2Σ−3YT3+YT2Σ−3RT3. |
Then,
˜A˜G=U[Ir000]UT+ϵU[0Σ−1RT3R3Σ−10]UT+ϵ∗U[0Σ−1YT3Y3Σ−10]UT+ϵϵ∗U[N1N2N3N4]UT, |
where
{N1=−Σ−1RT3Y3Σ−1−Σ−1YT3R3Σ−1,N2=Σ−1ZT3−Σ−1RT1Σ−1YT3−Σ−1YT1Σ−1RT3,N3=Z3Σ−1−Y3Σ−1R1Σ−1−R3Σ−1Y1Σ−1,N4=Y3Σ−2RT3+R3Σ−2YT3. |
˜G˜A=V[Ir000]VT+ϵV[0Σ−1R2RT2Σ−10]VT+ϵ∗V[0Σ−1Y2YT2Σ−10]VT+ϵϵ∗V[P1P2P3P4]VT, |
where
{P1=−Σ−1Y2RT2Σ−1−Σ−1R2YT2Σ−1,P2=Σ−1Z2−Σ−1Y1Σ−1R2−Σ−1R1Σ−1Y2,P3=ZT2Σ−1−RT2Σ−1YT1Σ−1−YT2Σ−1RT1Σ−1,P4=YT2Σ−2R2+RT2Σ−2Y2. |
Then, ˜A˜G and ˜G˜A are symmetric.
Furthermore,
˜A˜G˜A=U[Σ000]VT+ϵU[00R30]VT+ϵ∗U[00Y30]VT+ϵϵ∗U[−Σ−1RT3Y3−Σ−1YT3R30Z3−Y3Σ−1R1−R3Σ−1Y10]VT+ϵU[R1R200]VT+ϵϵ∗U[Σ−1YT3R30Y3Σ−1R1Y3Σ−1R2]VT+ϵϵ∗U[Σ−1RT3Y30R3Σ−1Y1R3Σ−1Y2]VT+ϵ∗U[Y1Y200]VT+ϵϵ∗U[Z1Z200]VT=U[Σ000]VT+ϵU[R1R2R30]VT+ϵ∗U[Y1Y2Y30]VT+ϵϵ∗U[Z1Z2Z3Z4]VT=˜A,˜G˜A˜G=V[Σ−1000]UT+ϵV[00RT2Σ−20]UT+ϵ∗V[00YT2Σ−20]UT+ϵϵ∗V[−Σ−1Y2RT2Σ−2−Σ−1R2YT2Σ−20ZT2Σ−2−RT2Σ−1YT1Σ−2−YT2Σ−1RT1Σ−20]UT+ϵV[−Σ−1R1Σ−1Σ−2RT300]UT+ϵϵ∗V[Σ−1Y2RT2Σ−20−YT2Σ−1RT1Σ−1YT2Σ−3RT3]UT+ϵ∗V[−Σ−1Y1Σ−1Σ−2YT300]UT+ϵϵ∗V[Σ−1R2YT2Σ−20−RT2Σ−1YT1Σ−1RT2Σ−3YT3]UT+ϵϵ∗V[M1M200]UT=V[Σ−1000]UT+ϵV[−Σ−1R1Σ−1Σ−2RT3RT2Σ−20]UT+ϵ∗V[−Σ−1Y1Σ−1Σ−2YT3YT2Σ−20]UT+ϵϵ∗V[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+ϵ∗ˆA0, |
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 (Im−ˆAˆA†)ˆA0(In−ˆA†ˆA), we obtain
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=(U[000Im−r]UT−ϵU[0Σ−1RT3R3Σ−10]UT)×(U[Y1Y2Y30]VT+ϵU[Z1Z2Z3Z4]VT)×(V[000In−r]VT−ϵV[0Σ−1R2RT2Σ−10]VT)=ϵU[000Z4−R3Σ−1Y2−Y3Σ−1R2]VT. |
Therefore, if
Z4=R3Σ−1Y2+Y3Σ−1R2, |
then
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=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
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=U[000Y4]VT+ϵU[0−Σ−1RT3Y4−Y4RT2Σ−1Z4−R3Σ−1Y2−Y3Σ−1R2]VT. |
Hence, if
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=0, |
then Y4=0 and
Z4=R3Σ−1Y2+Y3Σ−1R2. |
(ⅲ)⇔(ⅳ): By [16], if ˆA† exists, then
(Im−A0A†0)A1(In−A†0A0)=0. |
Moreover, if
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=0, |
then substituting
ˆA=A0+ϵA1, ˆA0=A2+ϵA3 |
and
ˆA†=A†0+ϵ[−A†0A1A†0+(AT0A0)†AT1(Im−A0A†0)+(In−A†0A0)AT1(A0AT0)†] |
into
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=0 |
gives
(Im−A0A†0)A2(In−A†0A0)+ϵ[(Im−A0A†0)(A3−A2A†0A1−A1A†0A2)(In−A†0A0)−(Im−A0A†0)A2(In−A†0A0)AT1(A0AT0)†A0−A0(AT0A0)†AT1(Im−A0A†0)A2(In−A†0A0)]=0, |
which implies
(Im−A0A†0)A2(In−A†0A0)=0 |
and
(Im−A0A†0)(A3−A2A†0A1−A1A†0A2)(In−A†0A0)=0. |
Conversely, if
(Im−A0A†0)A1(In−A†0A0)=0, |
then by [16], ˆA† exists. Moreover, if
(Im−A0A†0)A2(In−A†0A0)=0 |
and
(Im−A0A†0)(A3−A2A†0A1−A1A†0A2)(In−A†0A0)=0, |
then it is not difficult to see that
(Im−ˆAˆA†)ˆA0(In−ˆA†ˆA)=0. |
(ⅳ)⇔(ⅴ): It follows directly from Lemma 1.2.
It remains to show that
˜G=ˆA†+ϵ∗[−ˆA†ˆA0ˆA†+(ˆATˆA)†ˆA0T(Im−ˆAˆA†)+(In−ˆA†ˆA)ˆA0T(ˆAˆAT)†]. |
By a direct calculation, we have
ˆA†=V[Σ−1000]UT+ϵV[−Σ−1R1Σ−1Σ−2RT3RT2Σ−20]UT, | (2.5) |
ˆA†ˆA0ˆA†=V[Σ−1Y1Σ−1000]UT+ϵV[Q1Σ−1Y1Σ−2RT3RT2Σ−2Y1Σ−10]UT, | (2.6) |
where
Q1=Σ−1(−R1Σ−1Y1Σ−1−Y1Σ−1R1Σ−1+Z1Σ−1+Σ−1RT3Y3Σ−1+Y2RT2Σ−2). |
(ˆATˆA)†ˆA0T(Im−ˆAˆA†)=V[0Σ−2YT300]UT+ϵV[−Σ−2YT3R3Σ−1Q20RT2Σ−3YT3]UT, |
where
Q2=Σ−2ZT3−Σ−2RT1Σ−1YT3−Σ−1R1Σ−2YT3−Σ−2YT1Σ−1RT3. |
(In−ˆA†ˆA)ˆA0T(ˆAˆAT)†=V[00YT2Σ−20]UT+ϵV[−Σ−1R2YT2Σ−20Q3YT2Σ−3RT3]UT, | (2.7) |
where
Q3=ZT2Σ−2−RT2Σ−1YT1Σ−2−YT2Σ−2R1Σ−1−YT2Σ−1RT1Σ−2. |
Now, it can be seen from (2.4)–(2.7) that
˜G=ˆA†+ϵ∗[−ˆA†ˆA0ˆA†+(ˆATˆA)†ˆA0T(Im−ˆAˆA†)+(In−ˆA†ˆA)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+ϵ∗A2+ϵϵ∗A3. |
Since
r[A1A0A00]=r[A2A0A00]=r[A3−A2A†0A1−A1A†0A2A0A00]=2=2r(A0), |
then by Theorem 2.1(ⅴ), the HDMPGI of ˜A exists.
A direct computation shows that
ˆA†=(A0+ϵA1)†=[120120]+ϵ[−112−1212] |
and
˜A†=ˆA†+ϵ∗[−ˆA†ˆA0ˆA†+(ˆATˆA)†ˆA0T(Im−ˆAˆA†)+(In−ˆA†ˆA)ˆA0T(ˆAˆAT)†]=[120120]+ϵ[−112−1212]+ϵ∗[0101]+ϵϵ∗[−52−52−32−32]. |
Qi et al. [28] introduced a total order ≤ over ˆR. Suppose
ˆp=p+ϵp0, ˆq=q+ϵq0∈ˆR. |
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+ϵ∗ˆp0, ˜q=ˆq+ϵ∗ˆq0∈˜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 ˜a≥0, then we call ˜a a nonnegative hyper-dual number.
Recall that for a dual vector
ˆx=x+ϵx0∈ˆRn, |
the Euclidean norm of ˆx is defined as [28]
‖ˆx‖={‖x‖+2ϵxTx0‖x‖,ifx≠0,‖x0‖ϵ,ifx=0. |
For a hyper-dual number ˜a, ‖˜a‖2 is also a hyper-dual number. We may study least-squares properties of HDMPGI by the total order. However, ‖˜a‖2 is not always nonnegative, for example,
‖ϵa1+ϵ∗a2+ϵϵ∗a3‖2=(ϵa1+ϵ∗a2+ϵϵ∗a3)T(ϵa1+ϵ∗a2+ϵϵ∗a3)=2ϵϵ∗aT1a2. |
For this reason, we introduce the following set:
˜R0m={a0+ϵa1+ϵ∗a2+ϵϵ∗a3∣a0,a1,a2,a3∈Rm,a0≠0ora0=0andaT1a2≥0}. |
For a hyper-dual vector
˜a=a0+ϵa1+ϵ∗a2+ϵϵ∗a3∈˜Rm,‖˜a‖2=˜aT˜a=‖a0‖2+2ϵaT0a1+2ϵ∗aT0a2+2ϵϵ∗(aT0a3+aT1a2). |
Hence, if ˜a∈˜R0m, then ‖˜a‖2≥0.
For ˜a∈˜R0m, we define the Euclidean norm of ˜a as follows:
‖˜a‖={‖a0‖+ϵaT0a1‖a0‖+ϵ∗aT0a2‖a0‖+ϵϵ∗(aT0a3+aT1a2‖a0‖−aT0a1aT0a2‖a0‖3),ifa0≠0,ϵ√aT1a2+ϵ∗√aT1a2+ϵϵ∗‖a3‖,ifa0=0,a1≠0,a2≠0andaT1a2≥0,ϵ‖a1‖+2ϵϵ∗aT1a3‖a1‖,ifa0=a2=0,a1≠0,ϵ∗‖a2‖+2ϵϵ∗aT2a3‖a2‖,ifa0=a1=0,a2≠0,ϵϵ∗‖a3‖,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=b1−A1x0,A0x2=b2−A2x0,A0x3=b3−A3x0−A2x1−A1x2. | (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
‖˜e‖2=‖˜u+˜v‖2=(˜u+˜v)T(˜u+˜v)=‖˜u‖2+‖˜v‖2+2˜uT˜v=‖˜u‖2+‖˜v‖2. | (3.4) |
Let
˜u=u0+ϵu1+ϵ∗u2+ϵϵ∗u3. |
Then,
‖˜u‖2=˜uT˜u=‖u0‖2+2ϵuT0u1+2ϵ∗uT0u2+2ϵϵ∗(uT0u3+uT1u2). | (3.5) |
If ˜u∈˜R0m, then it can be observed from (3.5) that
‖˜u‖2≥0, |
and thus
‖˜e‖2≥‖˜v‖2 |
by (3.4), and equality holds if and only if
‖˜u‖2=0. |
Let
˜e=e0+ϵe1+ϵ∗e2+ϵϵ∗e3, ˜v=v0+ϵv1+ϵ∗v2+ϵϵ∗v3. |
Then,
‖˜e‖2=‖e0‖2+2ϵeT0e1+2ϵ∗eT0e2+2ϵϵ∗(eT0e3+eT1e2), | (3.6) |
‖˜v‖2=‖v0‖2+2ϵvT0v1+2ϵ∗vT0v2+2ϵϵ∗(vT0v3+vT1v2). | (3.7) |
Since the system of real linear equations
A0x0=b0 |
is inconsistent, then e0≠0, and thus
‖˜e‖2>0. |
In this case, it follows from (3.1) that
‖˜e‖=‖e0‖+ϵeT0e1‖e0‖+ϵ∗eT0e2‖e0‖+ϵϵ∗(eT0e3+eT1e2‖e0‖−eT0e1eT0e2‖e0‖3). | (3.8) |
By the assumption, ˜v∈˜R0m, and then
‖˜v‖2≥0. |
We consider the following two cases:
Case 1. ‖˜v‖2>0. In this case, either v0≠0 or v0=0 and vT1v2>0. If v0=0 and vT1v2>0, then by (3.1),
‖˜v‖=ϵ√vT1v2+ϵ∗√vT1v2+ϵϵ∗‖v3‖. |
Hence, by (3.8), ‖˜e‖>‖˜v‖.
If v0≠0, then
‖˜v‖=‖v0‖+ϵvT0v1‖v0‖+ϵ∗vT0v2‖v0‖+ϵϵ∗(vT0v3+vT1v2‖v0‖−vT0v1vT0v2‖v0‖3). | (3.9) |
Subcase 1. ‖˜e‖2>‖˜v‖2.
In this case, by (3.6) and (3.7),
‖e0‖>‖v0‖or‖e0‖=‖v0‖, |
eT0e1>vT0v1or‖e0‖=‖v0‖, |
eT0e1=vT0v1,eT0e2>vT0v2or‖e0‖=‖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. ‖˜e‖2=‖˜v‖2.
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. ‖˜e‖2>‖˜v‖2=0.
By the assumption, ˜v∈˜R0m. If ‖˜v‖2=0, then by (3.7), v0=0 and
vT1v2=0. |
We need only to consider the following five subcases:
(ⅰ) v0=0, v1≠0, v2≠0, vT1v2=0. In this subcase, by (3.1),
‖˜v‖=ϵϵ∗‖v3‖. |
(ⅱ) v0=v1=0, v2≠0. In this subcase, by (3.1),
‖˜v‖=ϵ∗‖v2‖+2ϵϵ∗vT2v3‖v2‖. |
(ⅲ) v0=v2=0, v1≠0. In this subcase, by (3.1),
‖˜v‖=ϵ‖v1‖+2ϵϵ∗vT1v3‖v1‖. |
(ⅳ) v0=v1=v2=0. In this subcase, by (3.1),
‖˜v‖=ϵϵ∗‖v3‖. |
(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
‖˜u‖2≥0 |
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˜A†−In)˜b=[0−7.3]+ϵ[7.3−2.5]+ϵ∗[14.6−12.9]+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=[3.1−7.3]+ϵ[27.70.6]+ϵ∗[−5.7−6.7]+ϵϵ∗[23.833.4]. |
Therefore, by (3.1),
‖˜A˜A(1,3)˜b−˜b‖=‖˜A˜A†˜b−˜b‖=7.3+ϵ2.5+ϵ∗12.9−ϵϵ∗1.4 |
and
‖˜A˜x1−˜b‖=7.93+ϵ10.3+ϵ∗4−ϵϵ∗47. |
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[−2.5−2.8]+ϵ∗[11.6−6.8]+ϵϵ∗[24.6−32.2]. |
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∗[−51.8−51.8]∈˜R02 |
and
˜A˜x2−˜b=[0−7.3]+ϵ[−2.9−2.5]+ϵ∗[−16.8−12.9]+ϵϵ∗[−41.2−35.8]. |
It follows from (3.1) that
‖˜A˜x2−˜b‖=7.3+ϵ2.5+ϵ∗12.9+ϵϵ∗42.5. |
Hence,
‖˜A˜A(1,3)˜b−˜b‖<‖˜A˜x2−˜b‖. |
Corollary 3.1. Let ˆA∈ˆRm×n be such that ˆA† exists, ˆb∈ˆRm. Denote
ˆx0=ˆA(1,3)ˆb−(In−ˆA(1,3)ˆA)ˆw, |
where ˆw∈ˆRm is an arbitrary dual vector. Then,
‖ˆAˆx0−ˆb‖≤‖ˆAˆx−ˆb‖ |
for all ˆx∈ˆRn.
For a hyper-dual number
˜a=a0+ϵa1+ϵ∗a2+ϵϵ∗a3, |
if a0≠0, 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)˜h‖2=‖˜A†˜b‖2+‖(In−˜A†˜A)˜h‖2. | (3.10) |
If a hyper-dual vector ˜h satisfies
(In−˜A†˜A)˜h∈˜R0n, |
then
‖(In−˜A†˜A)˜h‖2≥0. |
Hence, it can be observed from (3.10) that
‖˜A†˜b+(In−˜A†˜A)˜h‖2≥‖˜A†˜b‖2. |
On the other hand, let
˜A=A0+ϵA1+ϵ∗A2+ϵϵ∗A3, ˜A†˜b+(In−˜A†˜A)˜h=x0+ϵx1+ϵ∗x2+ϵϵ∗x3. |
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 A0x0≠0. Hence, x0≠0 and ˜A†˜b+(In−˜A†˜A)˜h is appreciable. In this case,
‖˜A†˜b+(In−˜A†˜A)˜h‖2>0. |
Moreover, ˜A†˜b∈˜R0n implies
‖˜A†˜b‖2≥0. |
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 ˆA∈ˆRm×n be such that ˆA† exists, ˆb∈ˆRm. If ˆAˆA†ˆb is appreciable, then
‖ˆA†ˆb‖≤‖ˆA†ˆb+(In−ˆA†ˆA)ˆh‖ |
for all ˆh∈ˆRn.
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(n−1)+ϵnˆC(n−1), |
where ˆB(n−1) and ˆC(n−1) are two dual matrices of order n−1, and ϵn is a dual unit. Hence, a dual matrix of order n can be obtained by two dual matrices of order n−1. 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(n−1)+ϵnˆC(n−1)∈ˆRm×n(n). |
Then, ˆA(n) has a Moore-Penrose generalized inverse if and only if (ˆB(n−1))† exists and
[Im−ˆB(n−1)(ˆB(n−1))†]ˆC(n−1)[In−(ˆB(n−1))†ˆB(n−1)]=0. |
Moreover, if the Moore-Penrose generalized inverse of ˆA(n) exists, then
(ˆA(n))†=(ˆB(n−1))†+ϵnˆZ(n−1), |
where
ˆZ(n−1)=−(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†+[(ˆB(n−1))TˆB(n−1)]†(ˆC(n−1))T×[Im−ˆB(n−1)(ˆB(n−1))†]+[In−(ˆB(n−1))†ˆB(n−1)](ˆC(n−1))T[ˆB(n−1)(ˆB(n−1))T]†. |
Proof. If ˆA(n) has a Moore-Penrose generalized inverse, we may suppose that
ˆX(n)=ˆY(n−1)+ϵnˆZ(n−1) |
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(n−1)+ϵnˆC(n−1) |
and
ˆX(n)=ˆY(n−1)+ϵnˆZ(n−1) |
into the above four equations yields
ˆB(n−1)ˆY(n−1)ˆB(n−1)=ˆB(n−1), ˆY(n−1)ˆB(n−1)ˆY(n−1)=ˆY(n−1),(ˆB(n−1)ˆY(n−1))T=ˆB(n−1)ˆY(n−1), (ˆY(n−1)ˆB(n−1))T=ˆY(n−1)ˆB(n−1). |
Hence, the Moore-Penrose generalized inverse of ˆB(n−1) exists and
ˆY(n−1)=(ˆB(n−1))†. |
On the other hand, equating the dual parts of both sides of the equation
ˆA(n)ˆX(n)ˆA(n)=ˆA(n) |
gives
ˆC(n−1)=ˆC(n−1)(ˆB(n−1))†ˆB(n−1)+ˆB(n−1)(ˆB(n−1))†ˆC(n−1)+ˆB(n−1)ˆZ(n−1)ˆB(n−1), |
which is equivalent to
ˆB(n−1)ˆZ(n−1)ˆB(n−1)=ˆC(n−1)−ˆC(n−1)(ˆB(n−1))†ˆB(n−1)−ˆB(n−1)(ˆB(n−1))†ˆC(n−1)≜ˆD(n−1). |
Then,
ˆD(n−1)=ˆB(n−1)ˆZ(n−1)ˆB(n−1)=ˆB(n−1)(ˆB(n−1))†ˆB(n−1)ˆZ(n−1)ˆB(n−1)(ˆB(n−1))†ˆB(n−1)=ˆB(n−1)(ˆB(n−1))†ˆD(n−1)(ˆB(n−1))†ˆB(n−1)=−ˆB(n−1)(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†ˆB(n−1). |
Now we have
−ˆB(n−1)(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†ˆB(n−1)=ˆC(n−1)−ˆC(n−1)(ˆB(n−1))†ˆB(n−1)−ˆB(n−1)(ˆB(n−1))†ˆC(n−1), |
that is,
[Im−ˆB(n−1)(ˆB(n−1))†]ˆC(n−1)[In−(ˆB(n−1))†ˆB(n−1)]=0. |
Conversely, if (ˆB(n−1))† exists and
[Im−ˆB(n−1)(ˆB(n−1))†]ˆC(n−1)[In−(ˆB(n−1))†ˆB(n−1)]=0, |
then we will show that the Moore-Penrose generalized inverse of ˆA(n) exists, and the matrix
ˆX(n)=(ˆB(n−1))†+ϵnˆZ(n−1) |
is a Moore-Penrose generalized inverse of ˆA(n), where
ˆZ(n−1)=−(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†+[(ˆB(n−1))TˆB(n−1)]†(ˆC(n−1))T×[Im−ˆB(n−1)(ˆB(n−1))†]+[In−(ˆB(n−1))†ˆB(n−1)](ˆC(n−1))T[ˆB(n−1)(ˆB(n−1))T]†. |
Indeed, by checking the four Penrose equations, we have
ˆA(n)ˆX(n)ˆA(n)=(ˆB(n−1)+ϵnˆC(n−1))[(ˆB(n−1))†+ϵnˆZ(n−1)](ˆB(n−1)+ϵnˆC(n−1))=ˆB(n−1)+ϵn[ˆB(n−1)(ˆB(n−1))†ˆC(n−1)+ˆC(n−1)(ˆB(n−1))†ˆB(n−1)−ˆB(n−1)(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†ˆB(n−1)]. |
Note that the condition
[Im−ˆB(n−1)(ˆB(n−1))†]ˆC(n−1)[In−(ˆB(n−1))†ˆB(n−1)]=0 |
is equivalent to
ˆC(n−1)=ˆB(n−1)(ˆB(n−1))†ˆC(n−1)+ˆC(n−1)(ˆB(n−1))†ˆB(n−1)−ˆB(n−1)(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†ˆB(n−1), |
which means that
ˆA(n)ˆX(n)ˆA(n)=ˆA(n). |
Moreover,
ˆX(n)ˆA(n)ˆX(n)=(ˆB(n−1))†+ϵn{−(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†+(ˆB(n−1))†ˆB(n−1)[(ˆB(n−1))TˆB(n−1)]†(ˆC(n−1))T[Im−ˆB(n−1)(ˆB(n−1))†]+[In−(ˆB(n−1))†ˆB(n−1)](ˆC(n−1))TˆB(n−1)[(ˆB(n−1))T]†ˆB(n−1)(ˆB(n−1))†}. |
Notice that
(ˆB(n−1))†ˆB(n−1)[(ˆB(n−1))TˆB(n−1)]†=(ˆB(n−1))†ˆB(n−1)(ˆB(n−1))†[(ˆB(n−1))T]†=(ˆB(n−1))†[(ˆB(n−1))T]†=[(ˆB(n−1))TˆB(n−1)]† |
and
[ˆB(n−1)(ˆB(n−1))T]†ˆB(n−1)(ˆB(n−1))†=[(ˆB(n−1))T]†(ˆB(n−1))†ˆB(n−1)(ˆB(n−1))†=[(ˆB(n−1))T]†(ˆB(n−1))†=[ˆB(n−1)(ˆB(n−1))T]†. |
Therefore,
ˆX(n)ˆA(n)ˆX(n)=(ˆB(n−1))†+ϵnˆZ(n−1)=ˆX(n). |
Furthermore,
ˆA(n)ˆX(n)=ˆB(n−1)(ˆB(n−1))†+ϵn[ˆB(n−1)ˆZ(n−1)+ˆC(n−1)(ˆB(n−1))†]=ˆB(n−1)(ˆB(n−1))†+ϵn{[Im−ˆB(n−1)(ˆB(n−1))†]ˆC(n−1)(ˆB(n−1))†+[ˆC(n−1)(ˆB(n−1))†]T[Im−ˆB(n−1)(ˆB(n−1))†]} |
and
ˆX(n)ˆA(n)=(ˆB(n−1))†ˆB(n−1)+ϵn[(ˆB(n−1))†ˆC(n−1)+ˆZ(n−1)ˆB(n−1)]=(ˆB(n−1))†ˆB(n−1)+ϵn{(ˆB(n−1))†ˆC(n−1)[Im−(ˆB(n−1))†ˆB(n−1)]+[Im−(ˆB(n−1))†ˆB(n−1)][(ˆB(n−1))†ˆC(n−1)]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(n−1)+ϵnˆC(n−1) |
exists, then the Moore-Penrose generalized inverse of ˆB(n−1) exists, and the Moore-Penrose generalized inverse of ˆA(n) is of the form (ˆB(n−1))†+ϵnˆZ(n−1).
Let
ˆX(n)1=(ˆB(n−1))†+ϵnˆZ(n−1)1 |
and
ˆX(n)2=(ˆB(n−1))†+ϵnˆZ(n−1)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(n−1)1=ˆZ(n−1)2. |
Equating the dual part of both sides of the equality
ˆA(n)ˆX(n)1ˆA(n)=ˆA(n), |
we get
ˆC(n−1)=ˆB(n−1)(ˆB(n−1))†ˆC(n−1)+ˆB(n−1)ˆZ(n−1)1ˆB(n−1)+ˆC(n−1)(ˆB(n−1))†ˆB(n−1). | (4.1) |
Similarly, equating the dual part of both sides of the equality
ˆA(n)ˆX(n)2ˆA(n)=ˆA(n) |
gives
ˆC(n−1)=ˆB(n−1)(ˆB(n−1))†ˆC(n−1)+ˆB(n−1)ˆZ(n−1)2ˆB(n−1)+ˆC(n−1)(ˆB(n−1))†ˆB(n−1). | (4.2) |
Subtracting (4.1) from (4.2) gives
ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)ˆB(n−1)=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(n−1)1=(ˆB(n−1))†ˆB(n−1)ˆZ(n−1)1+(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†+ˆZ(n−1)1ˆB(n−1)(ˆB(n−1))† | (4.4) |
and
ˆZ(n−1)2=(ˆB(n−1))†ˆB(n−1)ˆZ(n−1)2+(ˆB(n−1))†ˆC(n−1)(ˆB(n−1))†+ˆZ(n−1)2ˆB(n−1)(ˆB(n−1))†. | (4.5) |
Then, by subtracting (4.4) from (4.5), we have
ˆZ(n−1)1−ˆZ(n−1)2=(ˆB(n−1))†ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)+(ˆZ(n−1)1−ˆZ(n−1)2)ˆB(n−1)(ˆB(n−1))†. | (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(n−1)ˆZ(n−1)1+ˆC(n−1)(ˆB(n−1))†]T=ˆB(n−1)ˆZ(n−1)1+ˆC(n−1)(ˆB(n−1))† |
and
[ˆB(n−1)ˆZ(n−1)2+ˆC(n−1)(ˆB(n−1))†]T=ˆB(n−1)ˆZ(n−1)2+ˆC(n−1)(ˆB(n−1))†. |
It follows that
ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)=[ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)]T=(ˆZ(n−1)1−ˆZ(n−1)2)T(ˆB(n−1))T=(ˆZ(n−1)1−ˆZ(n−1)2)T(ˆB(n−1))T[ˆB(n−1)(ˆB(n−1))†]T=(ˆZ(n−1)1−ˆZ(n−1)2)T(ˆB(n−1))TˆB(n−1)(ˆB(n−1))†=[ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)]TˆB(n−1)(ˆB(n−1))†=ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)ˆB(n−1)(ˆB(n−1))†. |
Now, it can be seen from (4.3) that
ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)=0. |
We can also obtain
(ˆZ(n−1)1−ˆZ(n−1)2)ˆB(n−1)=0 |
in a similar way. Substituting
ˆB(n−1)(ˆZ(n−1)1−ˆZ(n−1)2)=0 |
and
(ˆZ(n−1)1−ˆZ(n−1)2)ˆB(n−1)=0 |
into (4.6), we have
ˆZ(n−1)1=ˆZ(n−1)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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