On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx

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1 Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: P-ISSN: Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam. ssstat Pofesso of Mathematcs, asu Egeeg College,Kumbakoam. STRCT: The am of ths atcle s to toduce the oto of EP bmatces. Ths pape s coceed solely wth developg the popetes of EP bmatces. Real ad complex EP bmatces ae studed fo the ow heet popetes ad a umbe of geealzatos of the esults aalogous to EP matces have bee obtaed. KEYWORDS: EP bmatx, EP bmatx, Pemutato bmatx MS classfcato: 59, 55, 557. I. INTRODUCTION The cocept of a omal matx wth etes fom the complex feld was toduced 98 by O.Toepltz [6] who gave a ecessay ad suffcet codto that a complex matx to be omal. Sce, the may eseaches have developed the cocept ad may geealzatos of omalty wee studed. st, esults about EP complex matces, a cocept toduced by H.Schwedtfege [5] as a geealzato of omalty wee obtaed ad the [3,4] the oto of EP was exteded to matces ove abtay felds ad appled to obta esults about omal matces. I ths pape we toduce the oto of EP bmatces as a exteso of bmatces. complex matx of ode s called EP f the age spaces of ad ae equal. evlle [6] temed a EP Matx as a age hemta matx. II. ON EP IMTRICES Defto. be a bmatx. The the ull space of s defed by, N x x that s x x. Example ( say) Hee x such that x. Hece x N ( ). Defto.3 bmatx x s sad to be EP f t satsfes the codto x x (o) equvaletly N N. Defto.4 bmatx s sad to be of ak f both the compoets of,that s ad ae of same ak P a g e

2 Example ( say) Sce, both ad ae of ak 3 the ak of s 3. Remak.6 Thoughout ths pape we cosde bmatces whose compoets ae of same ak. Defto.7 x bmatx followg codtos: () has ak. Remak.8 () wth etes fom the complex feld f ad oly f That s f ad oly f Hee deote the th ow of the bmatx of thats both ad. O EP matces x s called EP f t satsfes the whee,,,..., ad deote the th colum Example.9 3 ( say) a a N( ),, a R a a Hee ak s, Sce ad ae of ak. lso the codto satsfed. Theefoe s a EP bmatx. Theoem. If s a bmatx the the followg statemets ae equvalet : () () () (v) s a x EP bmatx. has ak ad thee s a x bmatx N such that N. has ak ad thee s a o sgula x bmatx N such that ca be epeseted as, DD X P P X DX D X DD X DD X P P P P XD XD X X DX D X s N P a g e

3 O EP matces I D X P P X I I D I X P P X I whee P s a pemutato bmatx ad D s a x o sgula bmatx. f ad oly f (v) whee. To Pove: ( v) ( ) If pat: Suppose whee,,..., That s ad () t The t s ce by () Takg cojugate taspose o both sdes, t pplyg (v) we have Thus, Oly f pat:. Suppose whee,,,..., ad ( ) t s ce The 46 P a g e

4 O EP matces ( ) t = Takg cojugate taspose o both sdes, t pplyg ( v) we have Thus, Hece f ad oly f To pove ( ) ( v) If s a x be the EP bmatx. EP ad let the ows,, ad,, be lealy depedet. Hece Thus, ad so the colums,, ad,, ae lealy depedet. Sce the ak of ad ae, the sub bmatces D ad D fomed by the elemets the tesecto of ows,,... ad the colums,, of ad also the ows,, of espectvely, ae a x o sgula bmatx [5,P-5].,,... ad the colums Now, Pemultply ad postmultply ad by the Pemutato matx P, Pad P, P espectvely such that P P D P P = ad ca be wtte the fom, E D ad P P = D E D E D E = E Sce the fst block ow of ad ae of the same ak as ad ad thus thee s a ( ) bmatx H H H such that, H D E ad H D E ad hece by (), such that E D H E D ad H If D ad D have ak the D ad D So that D s ce D D D hold fo some 47 P a g e

5 O EP matces D D HD ad D D D D H D H D Whch s a cotadcto to the assumpto that the fst block colum of has ak. Wth H as above, let I I I Q H I H I H I It s vefed that D D Q Q Q P P Q ad Q Q Q P P Q = D D P Q Q P ad P Q Q P I I X D P P X I I ad I I X D P P X I I I I X P Set P P X X ad Q Q we have, D P X I I To pove v I D I X P P () X I I ssume that () s tue. I I I Set Q Q Q= X I X I X I D If Q Q, whee Q QQ s o sgula ad D s x o sgula bmatx the, Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Pe multply ad post multply by Q ad Q Q Q Q Q Q Q Q Q Q Q 48 P a g e

6 O EP matces Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ( D D) Q Q D D = Q Q Q Q DD D Q Q Q Q I s ce N N N N N = N, N ae osgula. whee D D D Q Q Q Q I N DD Q Q I D D ad N Q Q I ad Poof of To pove v N s evdet. deote the ull space of ad let N That s ad The N N N N N N N N N 49 P a g e

7 O EP matces N N N = ak N ut, sce ak dm N ak ) ak ) dm N dm N N N Thus, f the Lemma. ad Moe geeally, let N N Poof. be x EP bmatces. The R R ad R R y kow esult, thee s a bmatxc such that C C C The, f ad oly f N N be x bmatces of ak the R R f ad oly f N N ( Sce ak ak, we have N N Covesely, N N y kow esult, thee s a bmatx C C C C R R ut ak ak We have R R Lemma. C such that s ce N N f C ) ad be complex bmatces satsfyg. The N N Poof. 5 P a g e

8 O EP matces Lemma.3 v, xn ad y N x ad y (), v, x y v, x y ad N Poof v x y v, x y v, x y s ce v, x y v, s ceby v N N N N be complex bmatces, satsfyg. The v ad z N z v, z v, z v, z v, z s ce v, v, z N v N Lemma.4 ad be complex bmatces satsfyg. The N N Poof x N ad y N Now, y y (s ce y N ) y N Hece x, y x, y x cey N, (s ) x N 5 P a g e

9 Theoem.5 ak N N ak ( ) ad ak( ) ak ( ) E P ad s E P the s E P. Poof If the N N Theefoe, Sce ak( ) ak ), We have N N Smlaly N N The Sce s E P Sce s E P N N N Hece ak ( ) ak ut Thee foe, N N s EP. If ad O EP matces ae REERENCES []. d e Isael Thomas N.E.evlle, eaalzed Iveses Theoy ad pplcatos, Secod Edto. []. Ivg Jack Katz ad Mat H.Peal, o EP ad omal EP matces, [3]. Joual of Reseach of the Natoal ueau of stadads.mathematcs ad Mathematcal physcs, vol. 7, NO., Jauay Mach 966. [4]. M.Peal, o Nomal ad EP matces, Mchga Maths.J, 6, -5 (959) [5]. M.Peal, o Nomal EP matces, Mchga Math. J ad (96) [6]. H.Schwedtfege, Itoducto to lea algeba ad the Theoy of Matces. P.Noodhoff, oge, 95. [7]. Teopltz,O., Das lgebasche alage Zu Ee Satz Vo eje, Matu, Zetschft, (98), P a g e