On Spectral Theory Of K-n- Arithmetic Mean Idempotent Matrices On Posets

Transcription

1 Iteratioal Joural of Sciece, Egieerig ad echology Research (IJSER), Volume 5, Issue, February 016 O Spectral heory Of -- Arithmetic Mea Idempotet Matrices O Posets 1 Dr N Elumalai, ProfRMaikada, 3 Sythiya Abstract We Cosider idempotet ad -idempotet matrices ad correspodig examples are itroduced Let C -be the uitary space of order, C x the set of all complex x matrices, the fixed product disjoit traspositio is S ad S the set of all permutatio matrices o {1,,3,,} We defie idempotet ad k- idempotet matrices o spectral theory as a abstract geeralizatio of k- idempotet matrices o C x ad we defie arithmetic mea idempotet matrices Some of the most importat properties of - eige values are preseted i terms of -idempotet matrices Idex erms AMS Classificatio: 15A57, 11B5, 11C Matrices, idempotet (or) di-potet, idempotet ad k- idempotet matrices, - eige values ad spectral theory properties ad arithmetic mea idempotet matrices ad -arithmetic mea idempotet matrices o posets 1 Example: Example of x idempotet matrix is Example of 3x3 idempotet matrix is Defiitio: A Square matrix A is said to be idempotet matrix if A A for ay positive iteger I geeral, A P+1 A for ay positive iteger P Example : A 1 INRODUCION A -idempotet matrix is defied ad some of its basic characterizatio are derived, [] It is show that if A is a - idempotet matrix the it is quadripotet (ie, A 4 A) Necessary ad sufficiet coditio for the sum of two - idempotet matrices to be -idempotet, is determied ad the it is geeralized for the sum of idempotet matrices[1] A coditio for the product of two idempotet matrices to be - idempotet is also determied ad the it is geeralized for the product of idempotet matrices Relatio betwee power hermit a matrices (A*A ) ad idempotet matrices are ivestigated[3] It is proved that a idempotet matrices A reduces to a idempotet matrix whe it commutes with the associated permutatio matrix (ie, A A) ad also defied eige values ad eige vectors of a complex matrices as a specializatio of geeralized eige values problem Ax λbx here A A, A 3 A,, A h A 3 O -k idempotet matrices: Defiitio For a fixed product of disjoit traspositios S, a matrix A <aij> i C x is said to be idempotet if a a a k ( i) t tk( j) ij is equivalet to A A, Where t 1 is the associated permutatio matrix of A Example: idempotet matrices Defiitio: A idempotet matrix is a matrix i which whe multiplied by itself, yields itself ie, he matrix M is idempotet if ad oly if MMM For this Product MM to be defied, M must ecessarily be a Square matrix he, If A ISSN: All Rights Reserved 016 IJSER 409

2 Iteratioal Joural of Sciece, Egieerig ad echology Research (IJSER), Volume 5, Issue, February 016 Here A is a idempotet matrix with <1,4> <,3> he associated permutatio matrix is a matrix with oes o its south west orth east diagoal ad zeros everywhere else ie, (I-A) (I-A) I-A Hece, I-A is idempotet 4 idempotet matrix: Let, (A) It ca be easily verified that A A 31 Remark: A A implies that A A he followig relatios ca also be obtaied which would be useful i computatioal aspects A A A 3 A (or) A A 3 A 3 A 3 (or) A 3 (A) (or) (A) 3 heorem: Let A ad B two idempotet matrices he A + B is idempotet if ad oly if AB - BA A+B A + B (A +B ) (A+B) iff AB -BA 33 heorem: Let A ad B be -idempotet matrices If ABBA the AB is the also be -idempotet matrix AB (A ) (B ) A B AABB (AB) (by AB BA) Hece the matrix AB is - idempotet 34 heorem: Let A be a -idempotet matrix he I-A is -idempotet iff A is idempotet I-A (I-A) (I-A+A ) I A +A > -A +A+A0 > A-A 0> (A-A )0 > A is idempotet Coversely, if A is idempotet the A commutes with the permutatio matrix (cf lemma 5) (I-A) (I-A+A ) A Similarly for, (A) A It is a -k idempotet mea matrices 41 Eige Value: A -λi λ λ > λ 0,1 Eige values of mea ad A mea matrices are always same 5 Arithmetic mea idempotet matrices o posets: I this mea, it satisfies the coditio as follows A+B A+B 51 Example: A B A, B A B ISSN: All Rights Reserved 016 IJSER 410

3 Iteratioal Joural of Sciece, Egieerig ad echology Research (IJSER), Volume 5, Issue, February 016 A B A B 5 Arithmetic mea idempotet matrices o posets: I this case, it satisfies the coditios as follows 55 Example: A+B A+B If, A, B 3 A B A B A B A+B A+B 53 Example: If, A, B A B A B A B A B 56 heorem: Let A ad B be two Arithmetic mea idempotet matrices he A+B is Arithmetic mea idempotet matrices where A ad B A+B + A B iff AB - BA Hece, A+B A+B 54 Arithmetic mea of -k idempotet matrix o posets: I this case, it satisfies the coditio as follows (A+B) iff AB - BA Hece, the matrix A+B is arithmetic mea - idempotet matrix 57 heorem: Let A & B be Arithmetic Mea idempotet matrices If ABBA the AB is also be Arithmetic mea idempotet matrix, where A, B AB ISSN: All Rights Reserved 016 IJSER 411

4 Iteratioal Joural of Sciece, Egieerig ad echology Research (IJSER), Volume 5, Issue, February 016 (by ABBA) (AB) (AB) Hece, the matrix AB is arithmetic mea k-idempotet 58 heorem: If A is arithmetic mea -idempotet matrix the A, A, A are also arithmetic mea idempotet matrices, where A i) A A > > hus, A is arithmetic mea idempotet matrix ii) A A > A B hus, A is arithmetic mea idempotet matrix 59 Eige values of Arithmetic mea matrices: (A- λi) 0 > -λ (1-λ) 0 λ 0,1 1 λ 0 (B λi) 0 -λ λ > λ (1-λ) 0 λ 0, 1 is, 1 λ 0 0 Eige values of Arithmetic mea idempotet matrix λ I 0 A B - 0 A B 0 hus, A is arithmetic mea idempotet matrix iii) AA > > (½ λ) ¼ 0 (½-λ + ½ ) (1/ λ ½ ) 0 λ 0,1 Eige values of Arithmetic mea idempotet matrices are always same ISSN: All Rights Reserved 016 IJSER 41

5 Iteratioal Joural of Sciece, Egieerig ad echology Research (IJSER), Volume 5, Issue, February 016 REFERENCES [1] J Baksalary, ad O M Baksalary, Idempotecy of liear combiatios of two idempotet matrices, Liear Algebra Appl, pp 3-7, 31 (000) [] S rishamoorthy, ad Rajagopala, O - idempotet matrices, It RevPure Appl Math, pp 9701, 5(1) (009) [3] S rishamoorthy, ad Rajagopala, Spectral ad Spectral theory of idempotet matrices, It J Math Stat, pp 81-85, 7 (10), witer (010) [4] David A Harville, Matrix Algebra From a Statisticia s Perspective 1 Dr N Elumalai, Associative Professor,Departmet of mathematics,avccollege(autoomous),maampadal Mayiladuthurai-amil Nadu RMaikada,AsstProfessor, Departmet of mathematics, AVC College (Autoomous), Maampadal,Mayiladuthurai 3 Sythiya, II MSc, Mathematics, AVC College (Autoomous), Maampadal, Mayiladuthurai ISSN: All Rights Reserved 016 IJSER 413