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Construction of K – Involutory Matrices

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Construction of K – Involutory Matrices PROTEUS JOURNAL ISSN/eISSN: 0889-6348 Construction of k – Involutory Matrices 1Dr. R. Manikandan,2Dr. R. Anuradha & 3Ms. S. Sripriya 1Assistant Professor, 2Assistant Professor & 3Research scholar 1,2,3 Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, Tamil Nadu. ABSTRACT: The concept of K-involutary matrices are introduced. Some of characterization on K-involutary matrices are given and also discussed construction of k- involutary matrices. KEY WORDS: Idempotent matrix ,k idempotent, Involutary matrix, K- Involutary matrix. AMS CLASSIFICATIONS: 15A09, 15B05,15A99,15B99. I.INTRODUCTION A matrix ‘A’ that satisfies A2=I is called an involutary matrix. A theory for K-real and K-hermitian matrices, as a generalization of secondary real and secondary hermitian matrices was developed by Hill and Waters[3].Krishnamoorthy,Rajagopalan and vijayakumar [2] have studied the basic concepts of k-idempotent matrices as generalization of idempotent matrices .The secondary symmetric, skew symmetric and secondary orthogonal matrices have been analyzed by Ann Lee[1] in the year 1976.The secondary k-hermitian bimatrices are introduced in 2009 by Meenakshi,Krishnamoorthy and Ramesh[6]. In this paper the basic concepts of k-involutary matrices are introduced construction of involutary matrices was n discussed. Throughout the paper let C denotes the unitary space of order n and Cnxn be the T * -1 space of all complex nxn matrices. For a matrix A 휖 Cnxn, Ā , A ,A and A denote conjugate, transpose, conjugate transpose and inverse of the matrix A respectively. Let ‘K’ be a fixed product of disjoint transpositions in Sn the set of all permutation on {1,2, …….,n}. Hence it is involuntary(that is k2=identity permutation). Let ‘K’ be the associated permutation matrix of K .Clearly K satisfies the following properties: K=KT=K*=K,K2=I II. SOME DEFINITIONS AND EXAMPLES 2.1 Involutory matrix [4,5] VOLUME 11 ISSUE 9 2020 http://www.proteusresearch.org/ Page No: 589 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 A square matrix A is said to be involutory matrix, If A2= I 2.2 Example 4 −1 In 2X2 involutary matrix is 15 −4 2.3 Idempotent Matrix [1] A Symmetric matrix A is said to be a idempotent, if A2=A 2.4 Example 1 0 Example of 2X2 idempotent matrix is 0 1 2 −2 −4 Example of 3X3 idempotent matrix is −1 3 4 1 −2 −3 2.5 k- Involutory matrix A square matrix A is said to be involutory matrix, If kA2k= I 2.6 Example 3 −4 4 0 0 1 In 3X3 involutary matrix is A = 0 −1 0 , k = 0 1 0 −2 2 −3 1 0 0 III SOME OF THEOREMS Theorem: 3.1 Let A² and B² be two K-involutory matrices then A²+B² is K-involutory if and only if A² = −B² Proof: Let A² and B² be K-involutory matrices, Therefore, KIK = A² and KIK = B² Assume that, A² = −B² A²+B² = KIK+KIK VOLUME 11 ISSUE 9 2020 http://www.proteusresearch.org/ Page No: 590 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 = K(I+I)K = K(I)K A²+B² = KIK ∴ A²+B is K-involutory matrices. Conversely, assume that A²+B² is K-involutory matrices ∴ A²+B² = KIK = K(2I)K = K(I+I)K = KIK+KIK = A²+ B² This implies that A²= −B² Theorem: 3.2 If A1A2……….An and I1,I2 ,………. In are K-involutory belongs to a commuting family of matrices the ∏ 퐼ᵢ is K-involutory matrices. Proof: Let A1,A2,……….An be K-involutory matrices ∏ 퐼ᵢ = K (A1A2……….An. A1A2……….An) K = K (A1² A2²………. An²) K = KA1²K. KA2²K……….KAn²K = I1²,I2²,……….In² K(∏ 퐴ᵢ)K = ∏ 퐼ᵢ ∴ ∏ 퐼ᵢ is K-involutory matrices. Theorem: 3.3 Let A² and B² be K-involutory matrices. If (AB)² = (BA)². Then (AB)² is also K- involutory matrices. VOLUME 11 ISSUE 9 2020 http://www.proteusresearch.org/ Page No: 591 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 Proof: Let A² and B² be two K-involutory matrices. Since KIK = A² and KIK = B² Assume that (AB)² = (BA)² (AB)² = (BA)² = B²A² = KIK. KIK = KI.IK ∵ I²=I = KI²K = KIK (AB)² = KIK Hence the matrix (AB)² is K-involutory matrices. Theorem: 3.4 If I∈Cⁿˣⁿ be K-involutory matrices then IT is also K-involutory. Proof: IT = (KA²K)T = K (AA)T K = KATATK = K(AT)²K IT = K(AT)²K ∴ IT is K-involutory. Theorem:3.5 VOLUME 11 ISSUE 9 2020 http://www.proteusresearch.org/ Page No: 592 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 If I∈Cⁿˣⁿ be K-involutory matrices then i) I̅ is K-involutory ii) Iˉ¹ is K-involutory Proof: i) 퐾퐴²퐾 = K퐴²K = K(퐴̅)²K 퐾퐴²퐾 = I̅ ∴ I̅ is K-involutory. ii) (KA²K)ˉ¹ = K(A²)ˉ¹ K = K(Aˉ¹)²K (KA²K)ˉ¹ = Iˉ¹ ∴ Iˉ¹ is K-involutory. Theorem:3.6 If I∈Cⁿˣⁿ be a K-involutory matrices then In is K-involutory for all positive integers ‘n’. Proof: In = (KA²K)n = KA²K.KA²K……….n times = KA²nK In = K(A)²n K ∴ In is K-involutory. Theorem:3.6 If I∈Cⁿˣⁿ be K-involutory matrices then -I is K-involutory. Proof: Let K-involutory matrices KA²K = I VOLUME 11 ISSUE 9 2020 http://www.proteusresearch.org/ Page No: 593 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 - (KA²K) = (K-(A²)K) = (K(-A²)K) = -I ∴ - (KA²K) = -I Hence -I is K-involutory matrices. IV CONCLUSION The notion of k-Involutary matrices reflects the fact that it is a generalization of Involutary matrices. Using this concept, we may also apply these results to con k-Involutary matrices, s k-Involutary matrices etc. V REFERENCES [1] Erdos J.A: On products of idempotent matrices, Glasgow Math.J. 8(1967). [2] Krishnamoorthy S and P.S. Meenakshi: On sums and products of k-Tripotent matrices; Vol.2 issue 7 july 2013. [3] Hill and Waters S.R: On k-real and k-Hermitian matrices; journal comput; Vol.4,no.2,DEC(2008). [4] Hazra A.K, Matrix : Algebra,Calculus and Generalised Inverse, Part-1. [5] Hazra A.KMatrix : Algebra,Calculus and Generalised Inverse, Part-2, [6] Meenakshi. A.R, Krishnamoorthy. S and Ramesh. G: On Secondary k-Hermitian Matrices; J. Modernsci.,IC(2009). VOLUME 11 ISSUE 9 2020 http://www.proteusresearch.org/ Page No: 594.
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