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Research article

Antioxidant, α-glucosidase, antimicrobial activities chemical composition and in silico analysis of eucalyptus camaldulensis dehnh

  • Received: 21 April 2024 Revised: 23 June 2024 Accepted: 12 July 2024 Published: 22 July 2024
  • Throughout history, medicinal plants have been the primary source for preventing and treating infectious diseases and other health issues. The flowering plant Eucalyptus camaldulensis, also called river red gum, is a member of the Myrtaceae family and has numerous traditional uses. The objectives of the present study were to identify the essential oil components of Eucalyptus camaldulensis using Gas Chromatography Mass Spectrometry (GCMS), and to determine the antioxidant, antidiabetic, and antibacterial activities of ethanol, aqueous, and essential oil extracts from E. camaldulensis leaves. Addtionally, the essential oil constituents that were identified underwent an in silico analysis. The efficacy of various extracts in combat pathogens and free radicals was assessed through the utilization of the 1,1-Diphenyl-2-Picryl Hydrazyl (DPPH), ferric reducing antioxidant power (FRAP), α-glucosidase inhibition, and disk diffusion methods. In terms of radical scavenging, reducing power, and α-glucosidase inhibitory activity, the essential oil showed strong antioxidant activity at 84.01 %, 20.1 mmol/g, and 78.2 %, respectively. The essential oil showed a potent antimicrobial action against Staphylococcus aureus and Pseudomonas aeruginosa, with 12 and 14 mm inhibitions, respectively, which were higher than ampicillin's 9 and 6 mm inhibitions, respectively. The GCMS analysis showed that the following chemicals were the most common: cis-11-hexadecenal (10.2%), trans-13-octadecenoic acid (9.5%), and 6-Octadecenoic acid, methyl ester, (Z)-(8.8%). The α-glucosidase enzyme was targeted in a docking study to investigate the antidiabetic properties of the 42 phytochemicals found in the essential oil extract. The compound, namely 5.alpha.-Androstan-16-one, showed the highest binding affinities of −8.6 Kcal/mol during the docking screening of the 42 identified phytochemicals against α-glucosidase. These two compounds show potential as competitive α-glucosidase inhibitors. E. camaldulensis will be a particularly useful source to improve health and fight communicable and non-communicable diseases. Nonetheless, human evaluations of E. camaldulensis safety and effectiveness are necessary, and more well-planned clinical trials are needed to confirm our in vitro and in silico findings.

    Citation: Abdulrahaman Mahmoud Dogara, Ateeq Ahmed Al-Zahrani, Sarwan W. Bradosty, Saber W. Hamad, Shorsh Hussein Bapir, Talar K. Anwar. Antioxidant, α-glucosidase, antimicrobial activities chemical composition and in silico analysis of eucalyptus camaldulensis dehnh[J]. AIMS Biophysics, 2024, 11(3): 255-280. doi: 10.3934/biophy.2024015

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  • Throughout history, medicinal plants have been the primary source for preventing and treating infectious diseases and other health issues. The flowering plant Eucalyptus camaldulensis, also called river red gum, is a member of the Myrtaceae family and has numerous traditional uses. The objectives of the present study were to identify the essential oil components of Eucalyptus camaldulensis using Gas Chromatography Mass Spectrometry (GCMS), and to determine the antioxidant, antidiabetic, and antibacterial activities of ethanol, aqueous, and essential oil extracts from E. camaldulensis leaves. Addtionally, the essential oil constituents that were identified underwent an in silico analysis. The efficacy of various extracts in combat pathogens and free radicals was assessed through the utilization of the 1,1-Diphenyl-2-Picryl Hydrazyl (DPPH), ferric reducing antioxidant power (FRAP), α-glucosidase inhibition, and disk diffusion methods. In terms of radical scavenging, reducing power, and α-glucosidase inhibitory activity, the essential oil showed strong antioxidant activity at 84.01 %, 20.1 mmol/g, and 78.2 %, respectively. The essential oil showed a potent antimicrobial action against Staphylococcus aureus and Pseudomonas aeruginosa, with 12 and 14 mm inhibitions, respectively, which were higher than ampicillin's 9 and 6 mm inhibitions, respectively. The GCMS analysis showed that the following chemicals were the most common: cis-11-hexadecenal (10.2%), trans-13-octadecenoic acid (9.5%), and 6-Octadecenoic acid, methyl ester, (Z)-(8.8%). The α-glucosidase enzyme was targeted in a docking study to investigate the antidiabetic properties of the 42 phytochemicals found in the essential oil extract. The compound, namely 5.alpha.-Androstan-16-one, showed the highest binding affinities of −8.6 Kcal/mol during the docking screening of the 42 identified phytochemicals against α-glucosidase. These two compounds show potential as competitive α-glucosidase inhibitors. E. camaldulensis will be a particularly useful source to improve health and fight communicable and non-communicable diseases. Nonetheless, human evaluations of E. camaldulensis safety and effectiveness are necessary, and more well-planned clinical trials are needed to confirm our in vitro and in silico findings.



    Throughout this work, Cp×n denotes the set involving p×n matrices with complex entries, and, for ACp×n, rank(A) is its rank, A is its conjugate-transpose matrix, N(A) is its null space, and R(A) is its range. The index ind(A) of ACp×p is the smallest nonnegative integer k for which the equality rank(Ak)=rank(Ak+1) is satisfied. The symbol I denotes the identity matrix of adequate size. Standard notations PS and PS,T denote, respectively, the orthogonal projector onto a subspace S and a projector onto S along T when Cp is equal to the direct sum of the subspaces S and T.

    Several definitions and properties of generalized inverses which are upgraded in this research are given. The Moore-Penrose inverse of ACp×n is uniquely determined A=XCn×p as the solution to well-known Penrose equations [1]:

    A=AXA,X=XAX,AX=(AX),XA=(XA).

    If X satisfies only equation XAX=X, it is an outer inverse of A. The outer inverse of A which is uniquely determined by the null space S and the range T is labeled with A(2)T,S=XCn×p and satisfies

    XAX=X,N(X)=S,R(X)=T,

    where sr=rank(A) is the dimension of the subspace TCn, and ps is the dimension of the subspace SCp.

    The following notation will be used:

    Cp,n;k:={(A,W):ACp×n, WCn×p{0} and k=max{ind(AW),ind(WA)}}.

    The notion of the Drazin inverse was extended to rectangular matrices in [2]. For selected (A,W)Cp,n;k, the W-weighted Drazin inverse AD,W=XCp×n of A is uniquely determined by the matrix equations

    XWAWX=X,AWX=XWA,(AW)k+1XW=(AW)k.

    Especially, if p=n and W=I, AD,I:=AD reduces to the Drazin inverse of A. Further, for ind(A)=1, AD:=A# becomes the group inverse of A. Recall that [2]

    AD,W=A[(WA)D]2=[(AW)D]2A.

    The notion of the core-EP inverse, proposed in [3] for a square matrix, was generalized to a rectangular matrix in [4]. If (A,W)Cp,n;k, the W-weighted core-EP inverse of A is the unique solution =XCp×n to

    WAWX=PR(WA)k,R((AW)k)=R(X).

    In a special case p=n and W=I, becomes the core-EP inverse of A. According to original definitions in [5] and [6,7,8], it is important to note

    and

    Some useful characterizations and representations of the core-EP inverse are presented in [3,9,10,11,12,13,14]. In the case ind(A)=1, reduces to the core inverse =A#AA of A [15].

    The weak group inverse (WGI) was presented for a square matrix in [16] as an extension of the group inverse. The WGI is extended in [17] to a rectangular matrix and in [18] to Hilbert space operators. For (A,W)Cp,n;k, the W-weighted WGI (W-WGI) of A is the unique solution A,W=XCp×n of the system [17,18]

    and it is expressed by [17,18]

    When p=n and W=I, A,I:=A reduces to the WGI of A

    Remark that, for 1=ind(A), A=A#. Useful results about WGI were given in [17,18,19,20,21,22,23].

    The concept of the m-weak group inverse (m-WGI) was introduced in [24] as an extension of the WGI. Exactly, if mN, the m-WGI of ACn×n is the unique matrix Am=XCn×n such that [25]

    (1.1)

    Recall that

    Clearly, A1=A, and particularly A2=()3A2 becomes the generalized group (GG) inverse of A, established in [26]. It is interesting that, if ind(A)m, Am=AD. Various properties of m-WGI were presented in [24,25,27,28].

    Recent research about m-WGI as well as the fact that m-WGI is an important extension of the WGI, GG, Drazin inverse, and group inverse motivated us to further investigate this topic. The current popular trend in the research of generalized inverses consists in defining new generalized inverses that are based on suitable combinations of existing generalized inverses as well as in their application in solving appropriate systems of linear equations. Considering the system (1.1) for defining m-WGI, our first aim is to solve a system of matrix equations which is an extension of the system (1.1) from the square matrix case to an arbitrary case. Since the m-WGI is restricted to square matrices, our main goal is to extend this notion to W-m-WGI inverses on rectangular matrices. To solve a certain system of matrix equations on rectangular complex matrices, we extend the notions of m-WGI, W-WGI, and the W-weighted Drazin inverse by introducing a wider class of generalized inverses, termed as the W-weighted m-WGI (W-m-WGI) for a rectangular matrix. Particularly, an extension of the GG inverse on rectangular matrices is obtained. It is important to mention that we recover significant results for the W-weighted Drazin inverse in a particular case. A class of systems of linear equations is found that can be efficiently solved applying W-m-WGI. This results is an extension of known results about the W-weighted Drazin solution and the Drazin solution of exact linear systems.

    The global structure of the work is based on sections with the following content. Several characterizations for the W-m-WGI are proved in Section 2 without and with projectors. We develop important expressions for the W-m-WGI based on core–EP, Drazin, and Moore-Penorse inverses of proper matrices. As a consequence, we introduce the weighted version of GG inverse and give its properties. Limit and integral formulae for computing the W-m-WGI are part of Section 3. Section 4 investigates applications of the W-m-WGI in solving specific matrix equations. Numerical experiments are presented in Section 5.

    We introduce the W-weighted m-WGI on rectangular matrices as a class of generalized inverses that includes notions of the m-WGI and the W-weighted WGI.

    Theorem 2.1. If (A,W)Cp,n;k and mN, then X=(W)m+1(AW)m1A is the unique solution to the matrix system

    (2.1)

    Proof. Using the identity AWW=, the subsequent transformations are obtained for X:=(W)m+1(AW)m1A:

    which further leads to

    Hence, X=(W)m+1(AW)m1A is a solution to (2.1).

    An arbitrary solution X to the system (2.1) satisfies

    which leads to the conclusion that X=(W)m+1(AW)m1A is the unique solution to (2.1).

    Definition 2.1. Under such conditions (A,W)Cp,n;k and mN, the W-weighted m-WGI (shortly W-m-WGI) inverse of A is defined by the expression

    Several special appearance forms of the W-m-WGI show its importance and are listed as follows:

    when p=n and W=I, the I-m-WGI coincides with the m-WGI Am=()m+1Am;

    if m=1, then (W)2A=A,W, that is, the W-1-WGI reduces to the W-WGI;

    for m=2, the W-2-WGI is introduced as A2,W=(W)3AWA and presents an extension of the GG inverse;

    in the case km, it follows Am,W=AD,W (see Lemma 2.1).

    Some computationally useful representations of the W-m-WGI are developed in subsequent statements.

    Lemma 2.1. If (A,W)Cp,n;k, mN and lk, then

    Furthermore, for mk, it follows that Am,W=AD,W.

    Proof. First, by induction on m, notice that gives

    Further, based on

    and

    (WA)k[(WA)k]=PR((WA)k)=PR((WA)l)=(WA)l[(WA)l],

    we obtain

    In the case mk, it follows that

    Am,W=A[(WA)D]m+2(WA)k[(WA)k](WA)m=A[(WA)D]m+2(WA)m=A[(WA)D]2=AD,W.

    Remark 2.1. Note that Am,W=(WA)m implies the interesting identity WAm,W=PR((WA)l)(WA)m for lk. This last identity is an extension of the classical property of the W-weighted Drazin inverse WAD,W=(WA)D. About the dual property AD,WW=(AW)D, if the equality A is satisfied (which is not true in general; for details see [5]), we can verify that Am,W=(AW)mA and so Am,WW=(AW)mAW.

    Representations for the W-2-WGI and W-weighted Drazin inverse are obtained as consequences of Lemma 2.1 when m=2 or m=lk, respectively.

    Corollary 2.1. If (A,W)Cp,n;k and lk, then

    and

    Notice that Corollary 2.1 recovers the known expressions for the W-weighted Drazin inverse [29,30].

    In Lemma 2.2, we show that the W-m-WGI Am,W is an outer inverse of WAW and find its range and null spaces.

    Lemma 2.2. If (A,W)Cp,n;k and mN, the following representations are valid:

    (i) Am,W=(WAW)(2)R((AW)k),N([(WA)k](WA)m);

    (ii) WAWAm,W=PR((WA)k),N([(WA)k](WA)m);

    (iii) Am,WWAW=PR((AW)k),N([(WA)k](WA)m+1W).

    Proof. (ⅰ) Based on Lemma 2.1 it follows that Am,W=A(WA)k[(WA)k+m+2](WA)m, which yields R(Am,W)R((AW)k) and

    Am,WWAWAm,W=A(WA)k[(WA)k+m+2](WA)k+m+2[(WA)k+m+2](WA)m=A(WA)k[(WA)k+m+2](WA)m=Am,W.

    Another application of Lemma 2.1 yields

    (AW)k=[(AW)D]m+2(AW)k+m+2=A[(WA)D]m+2(WA)k+m+1W=A[(WA)D]m+2(WA)k[(WA)k](WA)k+m+1W=Am,W(WA)k+1W (2.2)

    and so R((AW)k)R(Am,W). Thus, R(Am,W)=R((AW)k). Also,

    N([(WA)k](WA)m)=N([(WA)k+m+2](WA)m)=N([(WA)k+m+2](WA)m)=N(A(WA)k[(WA)k+m+2](WA)m)=N(Am,W).

    (ⅱ) By the part (ⅰ), WAWAm,W is a projector, and

    N(WAWAm,W)=N(Am,W)=N([(WA)k](WA)m).

    The equalities WAWAm,W=(WA)k+2[(WA)k+m+2](WA)m and

    (WA)k=(WA)k+2[(WA)D]2=(WA)k+2[(WA)k+2](WA)k+2[(WA)D]2=(WA)k+2PR([(WA)k+2])[(WA)D]2=(WA)k+2PR([(WA)m+k+2])[(WA)D]2=(WA)k+2[(WA)m+k+2](WA)m+k+2[(WA)D]2=WAWAm,W(WA)k+2[(WA)D]2=WAWAm,W(WA)k

    imply R(WAWAm,W)=R((WA)k).

    (ⅲ) It is clear, by (ⅰ), that R(Am,WWAW)=R(Am,W)=R((AW)k). The identity N(Am,WWAW)=N([(WA)k](WA)m+1W) is verified in a similar manner as in (ⅰ).

    Remark 2.2. For ACm×n, BCq×t, CCs×p, MCp×m, and NCn×q, by [31,32], the (M,N)-weighted (B,C)-inverse of A is represented by A(2,M,N)(B,C)=(MAN)(2)R(B),N(C). By Lemma 2.2, Am,W=(WAW)(2)R((AW)k),N([(WA)k](WA)m), and thus Am,W is the (W,W)-weighted ((AW)k),[(WA)k](WA)m)-inverse of A. Since the (B,C)-inverse of A is given as A(2)(B,C)=A(2)R(B),N(C) [33], it follows that Am,W=(WAW)(2)R((AW)k),N([(WA)k](WA)m) is the ((AW)k,[(WA)k](WA)m)-inverse of WAW.

    Lemma 2.2 and the Urquhart formula [1] give the next representations for Am,W.

    Corollary 2.2. If (A,W)Cp,n;k and mN, the W-m-WGI of A is represented as

    Am,W=(AW)k([(WA)k](WA)m+k+1W)[(WA)k](WA)m.

    If m=2 or m=k in Lemma 2.2 and Corollary 2.2, we obtain the next properties related to the W-2-WGI and W-weighted Drazin inverse.

    Corollary 2.3. If (A,W)Cp,n;k, the following statements hold:

    (i) A2,W=(WAW)(2)R((AW)k),N([(WA)k](WA)2)=(AW)k([(WA)k](WA)k+3W)[(WA)k](WA)2;

    (ii) WAWA2,W=PR((WA)k),N([(WA)k](WA)2);

    (iii) A2,WWAW=PR((AW)k),N([(WA)k](WA)3W);

    (iv) AD,W=(WAW)(2)R((AW)k),N((WA)k)=(AW)k([(WA)k](WA)2k+1W)[(WA)k](WA)k;

    (v) WAWAD,W=PR((WA)k),N((WA)k);

    (vi) AD,WWAW=PR((AW)k),N((AW)k).

    Some necessary and sufficient conditions for a rectangular matrix to be the W-m-WGI are considered.

    Theorem 2.2. If (A,W)Cp,n;k, XCp×n, and mN, the subsequent statements are equivalent:

    (i) X=Am,W;

    (ii) AWX=(W)m(AW)m1A and R(X)=R((AW)k);

    (iii) AWX=(W)m(AW)m1A and R(X)R((AW)k);

    (iv) AWXWX=X, X(WA)k+1W=(AW)k and [(WA)k](WA)m+1WX=[(WA)k](WA)m;

    (v) XWAWX=X, R(X)=R((AW)k) and [(WA)k](WA)m+1WX=[(WA)k](WA)m;

    (vi) XWAWX=X, AWX=(W)m(AW)m1A and XWA=(W)m+1(AW)mA;

    (vii) XWAWX=X, WAWX=W(W)m(AW)m1A and XWAW=(W)m+1(AW)m+1;

    (viii) XWAWX=X, AWXWA=(W)m(AW)mA, AWX=(W)m(AW)m1A and XWA=(W)m+1(AW)mA;

    (ix) XWAWX=X, WAWXWAW=W(W)m(AW)m+1, WAWX=W(W)m(AW)m1A and XWAW=(W)m+1(AW)m+1;

    (x) X=WAWX and AWX=(W)m(AW)m1A (or WAWX=W(W)m(AW)m1A);

    (xi) X=AWWX and WX=(W)m+1(AW)m1A;

    (xii) X=AD,WWAWX and AWX=(W)m(AW)m1A (or WAWX=W(W)m(AW)m1A);

    (xiii) X=(W)m+1(AW)m+1X and AWX=(W)m(AW)m1A (or WAWX=W(W)m(AW)m1A);

    (xiv) X=XW(W)m(AW)m1A and XWA=(W)m+1(AW)mA (or XWAW=(W)m+1(AW)m+1).

    Proof. (ⅰ) (ⅱ): It follows from Theorem 2.1 and Lemma 2.2.

    (ⅱ) (ⅲ): This implication is obvious.

    (ⅲ) (ⅰ): Because R(X)R((AW)k), we have

    X=(AW)kU=(AW)k[(AW)k](AW)kU=(AW)k[(AW)k]X,

    for some UCp×n. Notice that, by AWX=(W)m(AW)m1A,

    An application of Theorem 2.1 leads to the conclusion X=Am,W.

    (ⅰ) (ⅳ): For X=Am,W, Theorem 2.1 implies AWXWX=X. The equality (2.2) gives X(WA)k+1W=(AW)k. Using Lemma 2.1, we get X=A(WA)k[(WA)k+m+2](WA)m, which implies

    [(WA)k](WA)m+1WX=[(WA)k](WA)m+1WA(WA)k[(WA)k+m+2](WA)m=[(WA)k](WA)k+m+2[(WA)k+m+2](WA)m=[(WA)k](WA)k[(WA)k](WA)m=[(WA)k](WA)m.

    (ⅳ) (ⅴ): Notice, by X(WA)k+1W=(AW)k and

    X=AWXWX=(AW)2(XW)2X==(AW)r(XW)rX, (2.3)

    for arbitrary rN, that R(X)=R((AW)k). Hence, X=(AW)kU, for some UCp×n, and

    XWAWX=XWAW(AW)kU=(X(WA)k+1W)U=(AW)kU=X.

    (ⅴ) (ⅰ): The assumptions R(X)=R((AW)k) and XWAWX=X give (AW)k=XV=XWAW(XV)=XW(AW)k+1, for some VCn×p. Since X=(AW)kU, for some UCp×n, we get

    AWXWX=AWXW(AW)kU=AW(XW(AW)k+1)(AW)DU=AW(AW)k(AW)DU=(AW)kU=X.

    The assumption [(WA)k](WA)m+1WX=[(WA)k](WA)m yields

    (WA)k[(WA)k](WA)m+1WX=([(WA)k])[(WA)k](WA)m+1WX=([(WA)k])[(WA)k](WA)m=(WA)k[(WA)k](WA)m.

    Because (2.3) holds, we obtain

    Theorem 2.1 gives X=Am,W.

    (ⅰ) (ⅵ) (ⅶ): These implications are clear.

    (ⅶ) (ⅰ): Using XWAWX=X, WAWX=W(W)m(AW)m1A and XWAW=(W)m+1(AW)m+1, we get

    (ⅵ) (ⅷ) and (ⅶ) (ⅸ): These equivalences are evident.

    (ⅰ) (ⅹ): From X=(W)m+1(AW)m1A, we observe

    (ⅹ) (ⅰ): Applying X=WAWX and AWX=(W)m(AW)m1A, we obtain X=W(AWX)=W(W)m(AW)m1A=(W)m+1(AW)m1A.

    (v) (xiv): This equivalence follows as (ⅰ) (ⅶ).

    We also characterize the W-m-WGI in the following two ways.

    Theorem 2.3. If (A,W)Cp,n;k and mN, then

    (i) Am,W is the unique solution to

    WAWX=PR((WA)k),N([(WA)k](WA)m)andR(X)R((AW)k); (2.4)

    (ii) Am,W is the unique solution to

    XWAW=PR((AW)k),N([(WA)k](WA)m+1W)andR(X)R([(WA)m](WA)k). (2.5)

    Proof. (ⅰ) By Lemma 2.2, X=Am,W is a solution to (2.4). If the system (2.4) has two solutions X and X1, notice

    WAW(XX1)=PR((WA)k),N([(WA)k](WA)m)PR((WA)k),N([(WA)k](WA)m)=0

    and R(XX1)R((AW)k). Therefore,

    R(XX1)N(WAW)R((AW)k)N((AW)k)R((AW)k)={0},

    i.e., X=X1 is the unique solution of the system of Eqs (2.4).

    (ⅱ) Lemmas 2.1 and 2.2 imply validity of (2.5) for X=Am,W=A(WA)k[(WA)k+m+2](WA)m.

    The assumption that two solutions X and X1 satisfy (2.5) leads to the conclusion

    R(XX1)R([(WA)m](WA)k)N((WAW))R((WAWAm,W))N((WAWAm,W))={0},

    that is, X=X1.

    Corresponding characterizations of the W-2-WGI and W-weighted Drazin inverse are derived as particular cases m=2 and m=k of Theorem 2.3, respectively.

    Corollary 2.4. The following statements are valid for (A,W)Cp,n;k:

    (i) A2,W is the unique solution to

    WAWX=PR((WA)k),N([(WA)k](WA)2)andR(X)R((AW)k);

    (ii) A2,W is the unique solution to

    XWAW=PR((AW)k),N([(WA)k](WA)3W)andR(X)R([(WA)2](WA)k);

    (iii) AD,W is the unique solution to

    WAWX=PR((WA)k),N((WA)k)andR(X)R((AW)k);

    (iv) AD,W is the unique solution to

    XWAW=PR((AW)k),N((AW)k)andR(X)R([(WA)k]).

    Some formulae for the W-m-WGI are given in this section.

    We present a relation between a nonsingular bordered matrix and the W-m-WGI. Precisely, by Theorem 3.1, when the inverse of a proper bordered matrix is known, then the corresponding position of that inverse gives the W-m-WGI.

    Theorem 3.1. Let (A,W)Cp,n;k and mN. Assume that full-column rank matrices G and H fulfill

    N([(WA)k](WA)m)=R(G)andR((AW)k)=N(H).

    Then,

    N=[WAWGH0]

    is nonsingular, and

    N1=[Am,W(IAm,WWAW)HG(IWAWAm,W)G(WAWWAWAm,WWAW)H]. (3.1)

    Proof. Lemma 2.2 gives Am,W=(WAW)(2)R((AW)k),N([(WA)k](WA)m). Since

    R(IWAWAm,W)=N(WAWAm,W)=N(Am,W)=N([(WA)k](WA)m)=R(G)=R(GG)=N(IGG),

    we have (IGG)(IWAWAm,W)=0, that is, GG(IWAWAm,W)=(IWAWAm,W). From R(Am,W)=R((AW)k)=N(H), we get HAm,W=0. Set Y for the right hand side of (3.1). Then,

    NY=[WAWAm,W+GG(IWAWAm,W)WAW(IAm,WWAW)HGG(IWAWAm,W)WAWHHAmH(IAmWAW)H]=[WAWAm,W+IWAWAm,W(IWAWAm,W)WAWH(IWAWAm,W)WAWH0HH]=[I00I]=I.

    and similarly YN=I. So, N is nonsingular with N1=Y.

    We investigate limit and integral expressions for W-m-WGI motivated by limit and integral formulae of known generalized inverses [34,35,36,37].

    Theorem 3.2. If (A,W)Cp,n;k, mN, and lk, then

    Am,W=limλ0A(WA)l[(WA)l+m+2]((WA)l+m+2[(WA)l+m+2]+λI)1(WA)m=limλ0A(WA)l([(WA)l+m+2](WA)l+m+2+λI)1[(WA)l+m+2](WA)m.

    Proof. Lemma 2.1 gives Am,W=A(WA)l[(WA)l+m+2](WA)m. According to the limit representation for the Moore-Penrose inverse given in [36], we derive

    [(WA)l+m+2]=limλ0[(WA)l+m+2]((WA)l+m+2[(WA)l+m+2]+λI)1=limλ0([(WA)l+m+2](WA)l+m+2+λI)1[(WA)l+m+2],

    which implies the rest.

    Since W-m-WGI belongs to outer inverses, the limit representation of the outer inverse proposed in [35] implies the limit representation of the W-m-WGI.

    Theorem 3.3. Let (A,W)Cp,n;k and mN. Suppose that H1Cp×ss, R(H1)=R((AW)k), H2Cs×ns, and ns is the dimension of the subspace N(H2)=N([(WA)k](WA)m) in Cn. Then,

    Am=limv0H1(vI+H2WAWH1)1H2=limu0(uI+H1H2WAW)1H1H2=limv0H1H2(vI+WAWH1H2)1.

    Proof. Since Am,W=(WAW)(2)R((AW)k),N([(WA)k](WA)m), by [35, Theorem 7], it follows that

    Am,W=limu0H1(uI+H2WAWH1)1H2.

    Some integral formulae are established for the W-m-WGI.

    Theorem 3.4. If (A,W)Cp,n;k, mN, and lk, then

    Am,W=0A(WA)l[(WA)l+m+2]exp((WA)l+m+2[(WA)l+m+2]v)(WA)mdv.

    Proof. According to [34],

    [(WA)l+m+2]=0[(WA)l+m+2]exp((WA)l+m+2[(WA)l+m+2]v)dv.

    The proof is completed utilizing Am,W=A(WA)l[(WA)l+m+2](WA)m.

    Theorem 3.5. Let (A,W)Cp,n;k and mN. If HCp×n, R(H)=R((AW)k), and N(H)=N([(WA)k](WA)m), then

    Am=0exp[H(HWAWH)HWAWv]H(HWAWH)Hdv.

    Proof. Applying [37, Theorem 2.2], it follows that

    Am,W=(WAW)(2)R((AW)k),N([(WA)k](WA)m)=0exp[H(HWAWH)HWAWv]H(HWAWH)Hdv,

    which completes the proof.

    The W-m-WGI is applicable in studying solvability of some matrix and vector equations.

    In the case that ACm×n, xCn, and bCm, to find approximation solution to inconsistent system of linear equations Ax=b, a classical approach is to ask for, so called, generalized solutions, defined as solutions to GAx=Gb with respect to an appropriate matrix GCn×m [38]. It is important to mention that the system GAx=Gb is consistent in the case rank(GA)=rank(G). Such approach has been exploited extensively. One particular choice is G=A, which leads to widely used least-squares solutions obtained as solutions to the normal equation AAx=Ab. Another important choice is m=n, G=Ak, and k=ind(A), which leads to the so called Drazin normal equation Ak+1x=Akb and usage of the Drazin inverse solution ADb.

    Starting from the known equation WAWx=b, we use G=[(WA)k](WA)m to obtain the following equation (4.1).

    Theorem 4.1. If mN and (A,W)Cp,n;k, the general solution to

    [(WA)k](WA)m+1Wx=[(WA)k](WA)mb,bCn, (4.1)

    is of the form

    x=Am,Wb+(IAm,WWAW)u, (4.2)

    for arbitrary vector uCn.

    Proof. Let x be represented as in (4.2). Theorem 2.2 gives

    [(WA)k](WA)m+1WAm,W=[(WA)k](WA)m.

    So, x is a solution to (4.1) by

    [(WA)k](WA)m+1Wx=[(WA)k](WA)m+1WAm,Wb+[(WA)k](WA)m+1W(IAm,WWAW)u=[(WA)k](WA)mb.

    If Eq (4.1) has a solution x, based on

    Am,W=A[(WA)D]m+2(WA)k[(WA)k](WA)m,

    one concludes

    Am,Wb=A[(WA)D]m+2(WA)k[(WA)k](WA)mb=A[(WA)D]m+2([(WA)k])[(WA)k](WA)mb=A[(WA)D]m+2([(WA)k])[(WA)k](WA)m+1Wx=A[(WA)D]m+2(WA)k[(WA)k](WA)m+1Wx=Am,WWAWx,

    which yields

    x=Am,Wb+xAm,WWAWx=Am,Wb+(IAm,WWAW)x.

    Hence, x possesses the pattern (4.2).

    Choosing m=2 or mk in Theorem 4.1, we obtain the next result.

    Corollary 4.1. Let bCn and (A,W)Cp,n;k.

    (i) The general solution to

    [(WA)k](WA)3Wx=[(WA)k](WA)2b (4.3)

    possesses the form

    x=A2,Wb+(IA2,WWAW)u,

    for arbitrary uCn.

    (ii) If mk, the general solution to

    (WA)m+1Wx=(WA)mb (4.4)

    (or equivalently [(WA)k](WA)m+1Wx=[(WA)k](WA)mb)

    possesses the form

    x=AD,Wb+(IAD,WWAW)u,

    for arbitrary uCn.

    We study assumptions which ensure the uniqueness of the solution to Eq (4.1).

    Theorem 4.2. If mN and (A,W)Cp,n;k, x=Am,Wb is the unique solution to (4.1) in the space R((AW)k).

    Proof. Theorem 4.1 implies that (4.1) has a solution x=Am,WbR(Am,W)=R((AW)k).

    For two solutions x,x1R((AW)k) to (4.1), by Lemma 2.2, we obtain

    xx1R((AW)k)N([(WA)k](WA)m+1W)=R(Am,WWAW)N(Am,WWAW)={0}.

    Hence, the Eq (4.1) has uniquely determined solution x=Am,Wb in R((AW)k).

    Theorem 4.2 gives the next particular results.

    Corollary 4.2. Let bCn and (A,W)Cp,n;k.

    (i) x=A2,Wb is the unique solution in R((AW)k) to (4.3).

    (ii) x=AD,Wb is the unique solution in R((AW)k) to (4.4).

    Recall that, by [39], for WCn×p{0}, ACp×n, ind(AW)=k1, ind(WA)=k2, and bR((WA)k2), x=AD,Wb is the uniquely determined solution to

    WAWx=b,xR((AW)k1).

    Specifically, if ACn×n, W=I, ind(A)=k, and bR(Ak), x=ADb is the unique solution to [40]

    Ax=b,xR(Ak).

    For 1=ind(A) and bR(A), x=A#b is the uniquely determined solution to Ax=b. Notice that Theorem 4.2 and Corollary 4.2 recover the above mentioned results from [39] and [40].

    The identity (resp., zero) × matrix will be denoted by I (resp., 0). Denote by Dp, p1, the × matrix with its pth leading diagonal parallel filled by the entries of the vector 1={1,,1}Cp and 0 in all other positions.

    We perform numerical tests on the class of test matrices of index , given by

    {(CIC1I0C2Dp),  >0},  C,C1,C2C. (5.1)

    Example 5.1. The test matrix A in this example is derived using =4 and C=2,C1=3,C2=1 from the test set (5.1), and W is derived using =4 and C=1,C1=3/2,C2=4 from the test set (5.1). Our intention is to perform numerical experiments on integer matrices using exact computation. Appropriate matrices are

    A=(2I43I404D14)=(2000300002000300002000300002000300000100000000100000000100000000),W=(I43/2I4044D14)=(10003200001000320000100032000010003200000400000000400000000400000000).

    The matrices WA and AW fulfill k=ind(WA)=ind(AW)=2.

    (a) In the first part of this example, we calculate the Drazin inverse, the core-EP inverse, and W-m-WGI class of inverses based on their definitions. The Drazin inverse of WA is computed using

    (WA)D=(WA)2(WA5)(WA)2=(1/2I43/4I4+3/8D14+3/2D24+3/4D340404)=(120003438323401200034383200120003438000120003400000000000000000000000000000000),

    and the core-EP inverse of WA is equal to

    The W-weighted Drazin inverse of A is equal to

    AD,W=A[(WA)D]2=(WA)D

    and the W-weighted core-EP inverse of A is equal to

    The W-WGI (or W-1-WGI) inverse of A is given by

    the W-2-WGI inverse of A is equal to

    and for each mk the W-m-WGI inverse of A satisfies

    Additionally, Am=AD is checked for each mind(A).

    (b) Representations involved in Lemma 2.1 are verified using

    (c) In this part of the example, our goal is to verify results of Theorem 2.2.

    (c1) The statements involved in Theorem 2.2(ⅳ) are verified as follows.

    - In the case m=1 verification is confirmed by

    AWA,WWA,W=A,W;A,W(WA)3W=(4I46I4+24D14+12D24+48D340404)=(4000624124804000624120040006240004000600000000000000000000000000000000)=(AW)2;[(WA)2](WA)2WA,W=(8I412I4+6D140404)=(800012600080001260008000126000800012120001890061200945290246120362745291224612184527452)=[(WA)2]WA.

    - In the case m=2 results are confirmed by

    AWA2,WWA2,W=A2,W;A2,W(WA)3W=(AW)2;[(WA)2](WA)3WA2,W=2[(WA)2]WA=[(WA)2](WA)2.

    - Representations in the case m3 are confirmed by

    AWAm,WWAm,W=Am,W;Am,W(WA)3W=(AW)2;[(WA)2](WA)m+1WAm,W=m[(WA)2]WA=[(WA)2](WA)m.

    (c2) The statements involved in Theorem 2.2(ⅵ) are verified using verification of part (ⅳ) and the following computation.

    - In the case m=1

    - In the case m2 results are confirmed by

    Example 5.2. Consider A and W from Example 5.1 and the vector b=(22011201)T with intention to verify Theorem 4.1.

    In the case m=1 of (4.1), the general solution to [(WA)2](WA)2Wx=[(WA)2]WAb is equal to

    x1=A,Wb+(IA,WWAW)u,

    where u=(u1u2u3u4u5u6u7u8)T is a vector of unknown variables. Symbolic calculation gives

    x1=(3u526u63u712u8+523u626u73u8+523u726u8+3814(56u8)u5u6u7u8).

    Obtained vector x1 is verified using [(WA)2](WA)2Wx1=[(WA)2]WAb=(404062060901594292).

    In the case m2 of (4.1), the general solution to [(WA)2](WA)m+1Wx=[(WA)2](WA)mb is equal to

    xm=Am,Wb+(IAm,WWAW)u.

    Symbolic calculus produces

    xm=(3u526u63u712u8+1343u626u73u8+43u726u8+3814(56u8)u5u6u7u8).

    Correctness of the vector xm is verified using

    [(WA)2](WA)m+1Wx2=[(WA)2](WA)mb=(m1)(1041281240156270426609).

    In this research, we present an extension of the m-weak group inverse (or m-WGI) on the set of rectangular matrices, called the W-weighted m-WGI (or W-m-WGI). The W-m-WGI class presents a new, wider class of generalized inverses since this class involves the m-WGI, W-weighted weak group, and W-weighted Drazin inverse as special cases. Various characterizations and representations of W-m-WGI are developed. Usability of the W-m-WGI class in solving some constrained and unconstrained matrix equations and linear systems is considered. Some new properties of the weighted generalized group inverse and some known properties of the W-weighted Drazin inverse are obtained as corollaries. The given numerical examples confirm the obtained results.

    There is increasing interest in the investigation of the WGI and its generalizations, and so for further research in the near future, it may be interesting to consider its generalizations to Hilbert space operators or tensors, iterative methods for approximation of W-m-WGI, or recurrent neural network (RNN) models for computing W-m-WGI.

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

    Dijana Mosić and Predrag Stanimirović are supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, grant no. 451-03-65/2024-03/200124.

    Predrag Stanimirović acknowledges support from the Science Fund of the Republic of Serbia, (No. 7750185, Quantitative Automata Models: Fundamental Problems and Applications - QUAM).

    This work is supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075-15-2022-1121).

    P. S. Stanimirović is an editorial board member for Electronic Research Archive and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.



    Conflict of interest



    The authors declare no conflicts of interest.

    Author contributions



    Conceptualizations: AM Dogara. Experimental analysis: SW Bradosty, SW Hamad, SH Bapir, TK Anwar and AM Dogara. in-silico: AA Al-Zahrani. All authors contributed to the drafting of the initial draft and all authors read and approved the final draft.

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