AIJRFANS19-

American International Journal of Research in Formal, Applied & Natural Sciences Available online at http://www.iasir.net ISSN (Print): 2328-3777, ISSN (Online): 2328-3785, ISSN (CD-ROM): 2328-3793 AIJRFANS is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) ON THE s-INVERSE AND s-GENERALIZED INVERSE OF BIMATRICES 1 2 Dr. N. Elumalai, Mr. R. Manikandan, 3Ms. K. Abinaya Associate professor, 2Assistant Professor, 3M.Sc., Mathematics 1,2,3 Department of Mathematics, A.V.C. College (Autonomous), 1 Mannampandal-609 305, Tamilnadu, India. ABSTRACT: The concept of s-inverse and s-generalized inverse of bimatrices are introduced for singular and semi- singular bimatrices. The solution of homogeneous and non-homogeneous system of equations are studied. KEYWORDS: Bimatrix, s-Inverse Bimatrix, s-Generalized Inverse of Bimatrix, Singular, Semi-Singular, NonSingular Bimatrices. AMS Classification: 15A09, 15A15, 15A57. I. INTRODUCTION Let Rn×n denote the space of n×n real bimatrices. We deal with s-generalized inverse of bimatrices. We have to mingle with the bimatrices and s- permutation matrix[1]. In this paper, we describe s- generalized inverse of a square bimatrix, as the unique solution of a certain set of equation. Which is also satisfy the moorepenrose equation, The Moore-penrose inverse 𝑷† of P is the unique solution of the equations (𝒊)(𝑷𝑿𝑷 = 𝑷), (𝒊𝒊)(𝑿𝑷𝑿 = 𝑿), (𝒊𝒊𝒊)(𝑷𝑿)∗ = (𝑷𝑿), (𝒊𝒗)(𝑿𝑷)∗ = (𝑿𝑷). The concept of a generalized inverse was first introduced by Fredholm (1903) and the concept of an inverse of a singular matrix seems to have been first introduced by Moore in 1920. If 𝑃1 and 𝑃2 are any two matrices then the matrix 𝑷𝑩 = 𝑷𝟏 ∪ 𝑷𝟐 is said to be Bimatrix. II. s- INVERSES OF BIMATRICES Definition: 2.1: Let 𝑷𝑩 be a square bimatrix of order n. Then, 𝑷𝑩 is said to be Invertible. If there exists a square bimatrix 𝑸𝑩 of order n such that 𝑷𝑩 𝑸𝑩 = 𝑸𝑩 𝑷𝑩 = 𝑰𝑩 And 𝑸𝑩 is called the Inverse of 𝑷𝑩 and is denoted by 𝑷−𝟏 𝑩 . Definition: 2.2: A square matrix 𝑃𝐵 is said to be Singular. If the determinant value of both the components are zero. (That is, |𝐴1 | = 0 and |𝐴2 | = 0). Definition: 2.3: A square matrix 𝑃𝐵 is said to be Non-Singular. If the determinant value of both the components are non-zero. Definition: 2.4: A square matrix 𝑃𝐵 is said to be Semi-Singular. If the determinant value of either one of the component is zero. Properties of s-Inverses of Bimatrices: 2.5: Let 𝑃𝐵 and 𝑄𝐵 be the two bimatrices, then the following holds: 1. (𝑉𝐵 𝑃𝐵 𝑄𝐵 𝑉𝐵 )−1 = 𝑉𝐵−1 𝑄𝐵−1 𝑃𝐵−1 𝑉𝐵−1 2. (𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 )−1 =𝑉𝐵 𝑃𝐵 𝑉𝐵 3. (𝜆𝑉𝐵 𝑃𝐵 𝑉𝐵 )−1 =𝜆−1 𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 4. (𝑉𝐵𝑇 𝑃𝐵𝑇 𝑉𝐵𝑇 )−1 = (𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 )𝑇 Proof: 1. Let 𝑉𝐵 𝑃𝐵 𝑄𝐵 𝑉𝐵 = (𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )(𝑄1 ∪ 𝑄2 )(𝑉1 ∪ 𝑉2 ) 𝑉𝐵 𝑃𝐵 𝑄𝐵 𝑉𝐵 = (𝑉1 𝑃1 𝑄1 𝑉1 ) ∪ (𝑉2 𝑃2 𝑄2 𝑉2 ) AIJRFANS 19-206; © 2019, AIJRFANS All Rights Reserved Page 25 Elumalai et al., American International Journal of Research in Formal, Applied & Natural Sciences, 26(1), March-May 2019, pp. 25-27 [𝑉𝐵 𝑃𝐵 𝑄𝐵 𝑉𝐵 ]−1 = [(𝑉1 𝑃1 𝑄1 𝑉1 ) ∪ (𝑉2 𝑃2 𝑄2 𝑉2 )]−1 [∵ (𝑃𝑄𝑅𝑆)−1 = (𝑆 −1 𝑅−1 𝑄 −1 𝑃 −1 )] = (𝑉1 𝑃1 𝑄1 𝑉1 )−𝟏 ∪ (𝑉2 𝑃2 𝑄2 𝑉2 )−𝟏 −1 −1 −1 −1 −1 −1 −1 −1 = (𝑉1 𝑄1 𝑃1 𝑉1 ) ∪ (𝑉2 𝑄2 𝑃2 𝑉2 ) = (𝑉1−1 ∪ 𝑉2−1 )(𝑄1−1 ∪ 𝑄2−1 )(𝑃1−1 ∪ 𝑃2−1 )(𝑉1−1 ∪ 𝑉2−1 ) ∴ (𝑉𝐵 𝑃𝐵 𝑄𝐵 𝑉𝐵 )−1 = 𝑉𝐵−1 𝑄𝐵−1 𝑃𝐵−1 𝑉𝐵−1 2. Let𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 = (𝑉1−1 ∪ 𝑉2−1 )(𝑃1−1 ∪ 𝑃2−1 )(𝑉1−1 ∪ 𝑉2−1 ) 𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 = (𝑉1−1 𝑃1−1 𝑉1−1 ) ∪ (𝑉2−1 𝑃2−1 𝑉2−1 ) −1 (𝑉𝐵 𝑃𝐵−1 𝑉𝐵−1 )−1 = [(𝑉1−1 𝑃1−1 𝑉1−1 ) ∪ (𝑉2−1 𝑃2−1 𝑉2−1 )]−1 = (𝑉1−1 𝑃1−1 𝑉1−1 )−1 ∪ (𝑉2−1 𝑃2−1 𝑉2−1 )−1 [∵(𝑃 −1 )−1 = 𝑃] ) ) (𝑉 (𝑉 = 1 𝑃1 𝑉1 ∪ 2 𝑃2 𝑉2 = (𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )(𝑉1 ∪ 𝑉2 ) ∴ (𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 )−1 = 𝑉𝐵 𝑃𝐵 𝑉𝐵 3. Let 𝜆𝑉𝐵 𝑃𝐵 𝑉𝐵 = λ[(𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )(𝑉1 ∪ 𝑉2 )] = λ[(𝑉1 𝑃1 𝑉1 ) ∪ (𝑉2 𝑃2 𝑉2 )] −1 −1 (𝜆𝑉𝐵 𝑃𝐵 𝑉𝐵 ) = [λ[(𝑉1 𝑃1 𝑉1 ) ∪ (𝑉2 𝑃2 𝑉2 )]] = 𝜆−1 [(𝑉1 𝑃1 𝑉1 )−1 ∪ (𝑉2 𝑃2 𝑉2 )−1 ] = 𝜆−1 [(𝑉1−1 𝑃1−1 𝑉1−1 ) ∪ (𝑉2−1 𝑃2−1 𝑉2−1 )] = 𝜆−1 (𝑉1−1 ∪ 𝑉2−1 )(𝑃1−1 ∪ 𝑃2−1 )(𝑉1−1 ∪ 𝑉2−1 ) −1 ∴ (𝜆𝑉𝐵 𝑃𝐵 𝑉𝐵 ) = 𝜆−1 𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 4. Let 𝑉𝐵𝑇 𝑃𝐵𝑇 𝑉𝐵𝑇 = (𝑉1𝑇 ∪ 𝑉2𝑇 )(𝑃1𝑇 ∪ 𝑃2𝑇 )(𝑉1𝑇 ∪ 𝑉2𝑇 ) 𝑉𝐵𝑇 𝑃𝐵𝑇 𝑉𝐵𝑇 = (𝑉1𝑇 𝑃1𝑇 𝑉1𝑇 ) ∪ (𝑉2𝑇 𝑃2𝑇 𝑉2𝑇 ) 𝑇 (𝑉𝐵 𝑃𝐵𝑇 𝑉𝐵𝑇 )−1 = [(𝑉1𝑇 𝑃1𝑇 𝑉1𝑇 ) ∪ (𝑉2𝑇 𝑃2𝑇 𝑉2𝑇 )]−1 = (𝑉1𝑇 𝑃1𝑇 𝑉1𝑇 )−1 ∪ (𝑉2𝑇 𝑃2𝑇 𝑉2𝑇 )−1 [∵ (𝑃𝑇 )−1 = (𝑃 −1 )𝑇 ] −1 −1 −1 𝑇 −1 −1 −1 𝑇 = (𝑉1 𝑃1 𝑉1 ) ∪ (𝑉2 𝑃2 𝑉2 ) = [(𝑉1−1 𝑃1−1 𝑉1−1 ) ∪ (𝑉2−1 𝑃2−1 𝑉2−1 )]𝑇 = [(𝑉1−1 ∪ 𝑉2−1 )(𝑃1−1 ∪ 𝑃2−1 )(𝑉1−1 ∪ 𝑉2−1 )]𝑇 𝑇 𝑇 𝑇 −1 ∴ (𝑉𝐵 𝑃𝐵 𝑉𝐵 ) = (𝑉𝐵−1 𝑃𝐵−1 𝑉𝐵−1 )𝑇 ∎ III. s-GENERALIZED INVERSES OF BIMATRICES Definition: 3.1: Moore-Penrose Inverse of a bimatrix 𝑃𝐵 is the unique solution of the following equations: 1. 𝑉𝐵 𝑃𝐵 𝑋𝐵 𝑃𝐵 = 𝑉𝐵 𝑃𝐵 2. 𝑉𝐵 𝑋𝐵 𝑃𝐵 𝑋𝐵 = 𝑉𝐵 𝑋𝐵 3. (𝑉𝐵 𝑃𝐵 𝑋𝐵 )∗ = 𝑉𝐵 𝑃𝐵 𝑋𝐵 4. (𝑉𝐵 𝑋𝐵 𝑃𝐵 )∗ = 𝑉𝐵 𝑃𝐵 𝑋𝐵 Lemma: 3.2: Let 𝑃𝐵 be an (𝑛 × 𝑛) real bimatrix. Then the following holds: 1. (𝑉𝐵 𝑃𝐵 )†† = 𝑉𝐵 𝑃𝐵 ∗ 2. ((𝑉𝐵 𝑃𝐵 )∗ )† = 𝑉𝐵† 𝑃𝐵† 3. If 𝑃𝐵 is a non-singular, then (𝑉𝐵 𝑃𝐵 )† = 𝑉𝐵−1 𝑃𝐵−1 4. (𝜆𝑉𝐵 𝑃𝐵 )† = 𝜆† 𝑉𝐵† 𝑃𝐵† ∗ ∗ 5. (𝑉𝐵∗ 𝑉𝐵 𝑃𝐵∗ 𝑃𝐵 )† = 𝑃𝐵† 𝑃𝐵† 𝑉𝐵† 𝑉𝐵† Proof: 1. Let (𝑉𝐵 𝑃𝐵 )† = [(𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )]† = [𝑉1 𝑃1 ∪ 𝑉2 𝑃2 ]† = (𝑉1 𝑃1 )† ∪ (𝑉2 𝑃2 )† † † [(𝑉𝐵 𝑃𝐵 ) ] = [(𝑉1 𝑃1 )† ∪ (𝑉2 𝑃2 )† ]† = (𝑉1 𝑃1 )†† ∪ (𝑉2 𝑃2 )†† = 𝑉1 𝑃1 ∪ 𝑉2 𝑃2 = (𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 ) ∴ (𝑉𝐵 𝑃𝐵 )†† = 𝑉𝐵 𝑃𝐵 2. Let (𝑉𝐵 𝑃𝐵 )∗ = [(𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )]∗ = [𝑉1 𝑃1 ∪ 𝑉2 𝑃2 ]∗ = (𝑉1 𝑃1 )∗ ∪ (𝑉2 𝑃2 )∗ [(𝑉𝐵 𝑃𝐵 )∗ ]† = [(𝑉1 𝑃1 )∗ ∪ (𝑉2 𝑃2 )∗ ]† = [(𝑉1 𝑃1 )∗ ]† ∪ [(𝑉2 𝑃2 )∗ ]† AIJRFANS 19-206; © 2019, AIJRFANS All Rights Reserved Page 26 Elumalai et al., American International Journal of Research in Formal, Applied & Natural Sciences, 26(1), March-May 2019, pp. 25-27 = = = = [(𝑉1 𝑃1 )† ]∗ ∪ [(𝑉2 𝑃2 )† ]∗ [(𝑉1 𝑃1 )† ∪ (𝑉2 𝑃2 )† ]∗ [(𝑉1 𝑃1 ∪ 𝑉2 𝑃2 )† ]∗ [∵(𝐴∗ )† = (𝐴† )∗ ] † ∗ [((𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )) ] ∗ ∴ ((𝑉𝐵 𝑃𝐵 )∗ )† = 𝑉𝐵† 𝑃𝐵† 3. Given 𝑃𝐵 is non-singular bimatrix⇒ both 𝑃1 and 𝑃2 are non-singular matrices. † (𝑉1 𝑃1 )† = ((𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )) † = (𝑉1 𝑃1 ∪ 𝑉2 𝑃2 ) = (𝑉1 𝑃1 )† ∪ (𝑉2 𝑃2 )† = (𝑉1 𝑃1 )−1 ∪ (𝑉2 𝑃2 )−1 = (𝑉1 𝑃1 ∪ 𝑉2 𝑃2 )−1 −1 = ((𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )) ∴ (𝑉𝐵 𝑃𝐵 )† = 𝑉𝐵−1 𝑃𝐵−1 † 4. Let (𝜆𝑉𝐵 𝑃𝐵 )† = (𝜆(𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )) = [𝜆(𝑉1 𝑃1 ∪ 𝑉2 𝑃2 )]† † † 5. = (𝜆(𝑉1 𝑃1 )) ∪ (𝜆(𝑉2 𝑃2 )) = 𝜆† 𝑉1† 𝑃1† ∪ 𝜆† 𝑉2† 𝑃2† = 𝜆† [(𝑉1 𝑃1 )† ∪ (𝑉2 𝑃2 )† ] † = 𝜆† ((𝑉1 ∪ 𝑉2 )(𝑃1 ∪ 𝑃2 )) ∴ (𝜆𝑉𝐵 𝑃𝐵 )† = 𝜆† 𝑉𝐵† 𝑃𝐵† Let 𝑉𝐵∗ 𝑉𝐵 𝑃𝐵∗ 𝑃𝐵 = (𝑉1∗ ∪ 𝑉2∗ )(𝑉1 ∪ 𝑉2 )(𝑃1∗ ∪ 𝑃2∗ )(𝑃1 ∪ 𝑃2 ) = 𝑉1∗ 𝑉1 𝑃1∗ 𝑃1 ∪ 𝑉2∗ 𝑉2 𝑃2∗ 𝑃2 ∗ ∗ † (𝑉𝐵 𝑉𝐵 𝑃𝐵 𝑃𝐵 ) = (𝑉1∗ 𝑉1 𝑃1∗ 𝑃1 ∪ 𝑉2∗ 𝑉2 𝑃2∗ 𝑃2 )† = (𝑉1∗ 𝑉1 𝑃1∗ 𝑃1 )† ∪ (𝑉2∗ 𝑉2 𝑃2∗ 𝑃2 )† = (𝑉1∗ † 𝑉1† 𝑃1∗ † 𝑃1† ) ∪ (𝑉2∗ † 𝑉2† 𝑃2∗ † 𝑃2† ) = = = ∴ (𝑉𝐵∗ 𝑉𝐵 𝑃𝐵∗ 𝑃𝐵 )† = 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. ∗ ∗ ∗ ∗ (𝑉1† 𝑉1† 𝑃1† 𝑃1† ) ∪ (𝑉2† 𝑉2† 𝑃2† 𝑃2† ) ∗ ∗ ∗ ∗ (𝑉1† ∪ 𝑉2† )(𝑉1† ∪ 𝑉2† )(𝑃1† ∪ 𝑃2† )(𝑃1† ∪ 𝑃2† ) ∗ ∗ ∗ ∗ (𝑃1† ∪ 𝑃2† )(𝑃1† ∪ 𝑃2† )(𝑉1† ∪ 𝑉2† )(𝑉1† ∪ 𝑉2† ) ∗ ∗ 𝑃𝐵† 𝑃𝐵† 𝑉𝐵† 𝑉𝐵† REFERENCES [∵(𝐴1 𝐵1 )† = 𝐴1† 𝐴†2 ] [∵(𝐴∗ 𝐴)† = 𝐴† (𝐴† )∗ ] ■ Adi Ben-Isreal and Thomos N.E Greville, ‘Generalized inverses: Theory and Application’ 2nd Ed. Network, NY:springer 2003. Awni M.Abu-Saman,’Solution of linearly independent equations by generalized inverse of a matrix’,intJ.sci Emerging Tech,vol-4 No.2 August 2012 pp: 138-142. Hartwig. R.E, Katz.I.J, ‘On products of EP matrices’ Linear Algebra Appl. 252(1997)338-345. Moore.E.H,‘Generalizedanalysis’,Philadelphia, American Philosophical society,1935. Moore. E.H, ‘On the Reciprocal of the general algebraic Matrix,’ Bull amer Math. soc., vol. 26(1920), PP(394-395) Penrose. R, ‘A Generalized inverse of matrices’ proc Cambridgephilos.soc. vol 51(1955), PP(406-413). Ramesh. G, Anbarasi. N, ‘On Bihermitian matrices’ IOSR journal of Mathematics, vol 9 Issue 6 (Jan 2014). Ramesh. G, Anbarasi.N, ‘On EPr Bimtarices’ Int. jour. of Mathematics and Statistics Invention (IJMSI) vol. 2 Issue 5 May 2014, PP(44-52). Rao. C.R. and Maitra. S.K, ‘Generalized inverse of matrices and its applications’ wiley New york,(1971). Shizhencheng, yongge Tian, ‘Two sets of new characterizations for normal and EP matrices’ Elsevier, Linear Algebra and its applications 375 (2003) (181-195). AIJRFANS 19-206; © 2019, AIJRFANS All Rights Reserved Page 27