The Diagonal Equivalence of A Non-Negative Quaternion Matrix to A Doubly Stochastic Matrix

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1 ISSN(Olie): ISSN (Prit): Egieerig echology Vol. 8, Issue, Jauary 09 he Diagoal Equivalece of A No-Negative Quaterio Matrix to A Doubly Stochastic Matrix Dr.Guasekara K [], Seethadevi R []. Departmet of Mathematics, Govermet Arts College (Autoomous), Kumbakoam, amiladu, Idia ABSRAC: Let A be a positive square quaterio matrix. he there exists two diagoal matrices D,D whose diagoal elemets are positive D AD is quaterio doubly stochastic matrices. Marcus Newma [6] maxfield mic [7] also studied their problem. Perfect mirsky have show that give a fully idecomposable matrix B, there exists a doubly stochastic o-egative matrix with the same zero patter. theory. Morishima [] hompso [3] have studied such operators i extedig the theorems of perro frobeius I this last two sectios of the paper the diagoal equivalece of a o-egative quaterio matrix to a doubly stochastic matrix. We cosider the diagoal equivalece of a o-egative quaterio matrix to a row doubly stochastic matrix. Give ay square o-egative matrix A with atleast oe positive elemet i each row it is clear that there exists a row doubly stochastic matrix C havig zeros i exactly the same positio as A. Recetly, sikhor [] proved the existece of D sure that DAD is row stochastic uder the assumptio A is a positive matrix. KEYWORDS: quaterio doubly stochastic matrices, fully idecomposable matrices, Frobeius-Koig theorem, eige values. FULLY INDECOMPOSABLE MARICES. Notatio:. INRODUCION Let A a st be a m quaterio doubly stochastic matrix. If st A 0 ; ifast 0, the we write A 0 ; if A 0 but A 0, the we write A 0.. Defiitio: Let A 0 be a a 0 for each s t the we write quaterio doubly stochastic matrix. he A is called reducible provided there exists a permutatio matrix P PAP has the form Copyright to IJIRSE DOI:0.5680/IJIRSE

2 ISSN(Olie): ISSN (Prit): Egieerig echology Vol. 8, Issue, Jauary 09 A A 0 B ---- (..) Where A A are o-empty square matrices. If A is ot reducible, it is irreducible. A matrix A 0 is called fully idecomposable. If there do ot exits permutatio matrices P Q PAQ has the form(..). By covetio every matrix is fully idecomposable if oly if sigle etry is positive. A 0 Lemma.3: A matrix A 0 is quaterio colum stochastic if all its (colum sums) are oe. he matrix is quaterio doubly stochastic provided it is both row stochastic & colum stochastic. Let A 0 be a quaterio doubly stochastic matrix. he A is fully idecomposable if oly is there are permutatio matrices P Q PAQ has a positive mai diagoal is irreducible. Let A be fully idecomposable. he by Frobeius-Koig theorem [7,P.97] there exists permutatio matrices P Q PAQ has a positive mai diagoal. But obviously PAQ is fully idecomposable thus irreducible. Coversely, suppose that C PAQ has a positive mai diagoal (positive mai diagoal) is irreducible. Sice C is fully idecomposable if oly if A is, we may assumec A. Suppose A is ot fully idecomposable, let P Qbe two permutatio matrices P AQ is of the form(..). suppose A has r colums, A P AP is agai a matrix with a positive mai A has -r colums. he we may write P AQ Q PQ AQ where diagoal is a permutatio matrix. But the it follows that Q permutes the first r colums of A amog themselves the last -r colums of A amog themselves. Hece A is of the form (..) A is reducible, which is cotradictio. heorem.4: If D D are diagoal matrices DAD ay permutatio matrices P Q we have D (PAQ)D S. Where D diagoal matrices. S is a quaterio doubly stochastic matrices, the for PDP D Q D Q are Let A is irreducible with positive mai diagoal, the by the Frobeius-Koig theorem [7,P.97) there exists permutatio matrices P Q PAQ has a positive mai diagoal. But obviously PAQ is irreducible with positive mai diagoal thus fully idecomposable. Copyright to IJIRSE DOI:0.5680/IJIRSE

3 ISSN(Olie): ISSN (Prit): Egieerig echology Vol. 8, Issue, Jauary 09 Coversely, suppose that C=PAQ with has a fully idecomposable. Sice C is irreducible positive mai diagoal, let P Qbe two permutatio matrices P AQ is of the form (..). Suppose A has r rows A has -r rows. he we may write P AQ AQ where A P AP is agai a matrix with a positive mai diagoal Q PQ is a permutatio matrix. But the it follows that Q permutes the first r rows of A amog themselves the last -r rows of A of amog themselves. Hece A is of the form (..) A D is fully idecomposable which is cotradictio. his proves the above lemma. Example: A 0, Sice A fully idecomposable. i j k 3 i j k 4 3j k A 3 i j k i j k 4 3k 4 3j k 4 3k 9 3j 5k 0 0 P Q PAQ he D AD heorems heorems 3.: C, C 0, i j k 3 i j k 4 3j k PAQ 3 i j k i j k 4 3k 4 3j k 4 3k 9 3j 5k Let A>0 be a fully idecomposable quaterio matrix. he there exist diagoal matrices D D with positive mai diagoals D AD is quaterio doubly stochastic. Moreover D D are uiquely determie up to scalar multiples. By(.4), E is eough to prove this theorem for a irreducible matrix A with a positive mai diagoal. let x 0 set D diag(x, x,..., x ) D diag(( x),...,( x) ). he D AD is quaterio doubly stochastic if oly if x x, where is the operator associated with A. Hece the results from Copyright to IJIRSE DOI:0.5680/IJIRSE

4 ISSN(Olie): ISSN (Prit): Egieerig echology Vol. 8, Issue, Jauary 09 Let A>0 be a irreducible matrix with a positive mai diagoal. Let be the operator associated with the matrix A. he is a eige value of with a uique eige vector u i P. Further more u>>0. heorem 3.: We ow show that the coditio that A 0 fully idecomposable is essetial ecessary for (3.) to hold. Let A 0 be a quaterio matrix. he there exist diagoal matrices D D D AD is quaterio doubly stochastic matrices if oly if after idepedet permutatios of rows colums A is the direct sum of fully idecomposable matrices. Suppose the matrix A obtaied from A by idepedet permutatios of rows colums is equal to () (K) (i) A... A where each A is fully idecomposable. By there exists diagoal matrices D...D () (K) D...D (i) (i) (i) D A D is quaterio doubly stochastic. If () (K) D D... D ; () (K) D D... D () (K) he D AD is quaterio doubly stochastic. Hece (.4 ) we ca fid D D quaterio doubly stochastic. Coversely, suppose that there exists D D DAD S is quaterio doubly stochastic D AD is S S S S... 0 SK S K... S K Where each Si is either fully idecomposable or a x zero matrix. But sice S is quaterio doubly stochastic, the row sums of S are all oe hece the colum sums of S beig less tha or equal to must i fact be equal to. Hece S 0,...,SK 0.Repeatig this argumet we obtai S S S... SK, where each S i is a quaterio doubly stochastic hece fully idecomposable. heorem 3.3: he Kroecker product of two quaterio fully idecomposable double stochastic matrices if oly if both are fully idecomposable doubly stochastic matrices. Copyright to IJIRSE DOI:0.5680/IJIRSE

5 ISSN(Olie): ISSN (Prit): Egieerig echology Vol. 8, Issue, Jauary 09 Suppose that A B is fully idecomposable, but A is ot fully idecomposable quaterio doubly stochastic matrices, the there exist permutatio matrices P Q A A PAQ 0 A ---- (3.3.) Where A A are as i (), Now (P I)(A B)(Q I) PAQ B A B A B 0 A B Where the Kroecker product of o-square matrices is defied i the obvious way sice (P I) (Q I) are permutatio matrices this gives a cotradictio quaterio doubly stochastic. Similarly, if B is ot fully idecomposable the there permutatio matrix R, where [R](i,k)( j,l)...s ils Kj, satisfies B A R(A B)R,from which the result follows as above. II. CONCLUSION I this paper, we ca discuss fully idecomposable doubly stochastic matrices to a diagoal equivalece of a o-egative quaterio matrix. REFERENCES. Brualdi.R.A, Permeat of the product of doubly stochastic matrices, proc.cambridge philos.soc.6(966), Brualdi.R.A, Kroecker products of fully idecomposable of ultrastrog digraphs, J.combiatorial theory, (967), Brualdi.R, Parter.S Scheider.H, he diagoal equivalece of a oegative matrix (to appear). 4. Dulmage.A.L. Medelsoh.N.S, he structure of powers of o-egative matrices, Caad.j.math.7 (965), Guasekara.K Seethadevi.R, Characterizatio heorems o Quaterio doubly stochastic matrices, JUSPS-A vol.9(4), (07) 6. Marcus.M Newma.M, upublished paper. 7. Marcus.M Mic.H, A survey of matrix theory matrix i equalities, Ally Bacos, Maxfield.J H. M, A doubly stochastic matrix equivalet to a give matrix, ices Amer. Math. Soc. 9 (96), Morishima.M, Geeralizatios of the frobeius-wielttheorems for o-egative square matrices. J.Lodo Math.Soc.36(96), Sikhor.R, A relatioship betwee arbitrary positive matrices doubly stochastic matrices A. Math. Statist. 35 (964, Sikhor.R Kopp.P,. Cocerig o-egative matrices doubly stochastic matrices. o apper. hompso.a.c, O the eige vectors of some ot-ecessarily liear trasformatios. Proc.Lodo, Mth.Soc. 5(965), Copyright to IJIRSE DOI:0.5680/IJIRSE