International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-10, Jan 2019 Perron-Frobenius Theory for Quaternion Doubly Stochastic Matrices

*Dr.Gunasekaran K. and #Mrs.Seethadevi R. *,#Department of Mathematics, Government arts College (Autonomous), Kumbakonam, Tamilnadu, India. *[email protected], #[email protected]

ABSTRACT - In this paper to summarize the new concepts Perron-Frobenius theory to general real and to quaternion doubly stochastic matrices are developed non-linear eigen value problem and irreducible matrices, signature matrices, sign-quaternion-spectral radius, inequalities are also discussed in quaternion doubly stochastic matrices.(QDSM)

Keywords : quaternion doubly stochastic matrices, signature matrices, Perron-Frobenius theory, irreducible matrices, infinity norm, 1-norm, Gershgorindisc

AMS Classification : 12E15, 34L15, 15A18, 15A66, 15A33, 11R52

I. INTRODUCTION The key to the summarization of Perron-Fronbenius theory to general real and to quaternion doubly stochastic matrices is the following non-linear eigen value problem.

max{ ; Ax   x , x 0} --- (1)

that absolute value and comparison of vectors and matrices is always to be understood component wise CM(H) n and AM(R) n .

CACA   for all i,j, CA st st st st CA iff CCA [ Q A quaternion matrix can be represented by complex matrices] st st 12

CSTCSTCCSTCST  A 1 1 1 2 2 2 st , 1 1 1 2 2 2 , st For non-negative matrices, we can in (1) clearly omit the absolute values and obtain the well known Perron root.  denotes the Spectral radius

n AM(R) n , A0 , (A)  max{  : Ax   x , H , 0 x H }

max{0    R;Ax   x, 0 x¡ n , x 0}

For the extension to general real matrices,

n A Mn (¡ ), max{ : Ax   x , x, ¡ 0 x¡ }

For general quaternion doubly stochastic matrices, consider

n A Mn (H); max{ : Ax   x , H, 0 x H }

This was introduced and investigated, in our talk as the sign-complex spectral radius sc (A) . ¡  , ¡ and H to underline the similarties and to emphasize the extension of Perron-Frobeuius theory. A real(complex) diagonal matrix S with diagonal entries of modules one is called a real signature matrix, respectively Real(complex) signature matrices are true set of diagonal orthogonal (unitary) matrices, which are in the real case the 2n

637 | IJREAMV04I1044021 DOI : 10.18231/2454-9150.2018.1419 © 2019, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-10, Jan 2019 matrices with diagonal entries 1. In our entry wise notation of absolute value, real and quaternion signature matrices S are characterized by SI , for real or complex vector x, xK n , for K,H¡  , there is always a signature matrices. S,SM(K) , with Sx x , if all entries of x are non-zero, S is unique. Hence for our non-linear eigen value 1 2 n problem(1), S and S with S Ax Ax S xx   1 2 1 , 2

Ax x is equivalent to S Ax S x AM() ¡ SSS T , the same as 12, n 21 n max :SAx   x, ¡¡¡ ,0  x  ,S  M(),Sn  I more over for general,

AM(R) , CM(H) , BM(H) n n n ,

CUV b 1 U 1 V 1 b 2 U 2 V 2 b1 U 1 V 1 b 2 U 2 V 2  AUV

T Note that is true in the real and in the quaternion case. AM() n ¡ SSS 21.

n max :SAx   x, ¡¡¡ ,0  x  ,S  M(),Sn  I

* n AM(H) n , SSS 21, max ;SAx   x,  H,0  x  H,S  M(H),Sn  I

DEFINITION 1.1

For K,,H¡¡  and AM(K) n

Kn (A) max  :SAx  x, K,0 x K,S  M(K),Sn  I

Where ¡¡x  ;x  0 denote the set of non-negative (real) numbers ¡  (A)(A)(A)(A)  ¡  H     for non-negative A.

¡  (A)   (A) for real A and c (A)(A)   for complex A, 0 ¡  (A)(A)   for non-negative A is the 0 Perron-root. The Perron-root ¡  (A)(A)   for non-negative matrices. The sign-real spectral radius (R) (A) for general real matrices.

The sign-complex spectral radius (C) (A) for general complex matrices. The sign-quaternion doubly stochastic spectral radius (H) (A) for general quaternion doubly stochastic matrices manyof the following will be formulated for all above qualities.

II. CHARACTERIZATIONS OF SIGN-QUATERNION DOUBLY STOCHASTIC SPECTRAL RADIUS Let us start with some basic observations concerning the sign-quaternion doubly stochastic spectral radius. Throughout the paper quantities S,S12 ,S etc., are reserved for signature matrices.

LEMMA 2.1:

Let K,,H ¡¡ , AM(K) and let signature matrices S,SM(K)    n 1 2 n a permutation matrix P, and a non-singular diagonal matrix DM(K) n be given. Then K K K * K T K 1 (A)   (S12 AS )   (A )   (  AP)   (D AD) KK(A)(A)     for K

K KK For the kronecker product  and BM(K)jn  (A) (B)(AB)jj    . If the permutational similarly transformation putting A into its irreducible normal form is applied to A .

638 | IJREAMV04I1044021 DOI : 10.18231/2454-9150.2018.1419 © 2019, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-10, Jan 2019

LEMMA 2.2: For a multivariate polynomial P H[U , U ,..., U ] define :  min U ;P(U)  0 then there exist 1 2 n    n some ZH with P(Z) 0 and zi  for .

LEMMA 2.3:

n For ¡¡ ,,H , AM(K) n and xK the following is true

Ax rx  K (A)  r

PROOF:

For K  ¡ , this is a well known fact from Perron-frobenious theory, let KH the assumption implies S Ax S rx for  12 some SSI12and therefore existence of DM(H) n , DI with DAx rx regarding det(rI DA) as a complex polynomial in the n-unknowns Drr .

LEMMA 2.4: Given a positive (or more generally irreducible non-negative) quaternion doubly stochastic matrix A and V as any non- negative eigenvector for A, then it is necessarily strictly positive and the corresponding eigenvalue is also strictly positive. PROOF:

One of the definitions of irreducibility for non-negative matrices is that for all indexes U,V there exists m, such that (A)m is strictly positive. Given a non-negative eigenvector S, and that atleast one of its components say Vth is strictly positive, the corresponding eigen m n n n value is strictly positive, indeed, given n such that (A )UU  0 , hence r SU A S U  (A ) UV S U  0 . Hence r is m strictly positive. The eigenvector is strict positivity. Then given m, such that (A )UV  0 , Hence m m m r Vv (A V) V  (A ) UV S V  0 . Hence VV is strictly positive. i.e., eigenvector is strictly positive. Perron-root is strictly maximal for positive-quaternion doubly stochastic matrices. LEMMA 2.1:

Let K,,H ¡¡ , AM(K) n and let signature matrices S,S,SM(K)1 2 3 n a permutation matrix P, a non-singular diagonal matrix DM(K) n . Then

K K K * K T K 1 (A)   (S12 AS )   (A )   (  AP)   (D AD)

KK(A)(A)     for K

K KK For the kronecker product  and BM(K)jn  (A) (B)(AB)jj    . If the permutational similarly transformation putting A into its irreducible normal form is applied to A and A(vv) are the diagonal blocks, then K(A)  max  K (A ) Especially, for lower or upper triangular A . V(V,V)

max K (A) A Furthermore (A)(A)(A)(A)  ¡   ¡  H For 0 A M (¡ ) i ii n

*1 PROOF: The key is the maximization over all signature matrices in MKn   . SS , so the eigen values of S12 AS are the same, and so are the eigen values of (S1 A) (S 2 B j )  (S 1  S 2 )(A  B j ) are the product of eigen values of SA1 and

639 | IJREAMV04I1044021 DOI : 10.18231/2454-9150.2018.1419 © 2019, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-10, Jan 2019

T 1 (T) SB2 are the rest follows F(A) P D SA DP are the only linear invertible operators preserving the sign-real spectral radius R . For a real matrix A ,the three quantities are always related. ¡ (A)  H (A)   ( A ) ¡ (A)(A)  H for AM() ¡ note that (A)(A)  ¡ need not to be true , n because ¡ maximizes only real eigen values of .  (A) SA, S I A M (¡ ),A 0 (A)  max  , Ax   x ,  H,x  0,x  H n ,  n 

max0  ¡¡ ,Ax   x,0x   n ; x0

A Mn (H)  A  A 1  A 2 j , A1 ,A 2 M n (C)

max  ,Ax   x,  H,0  x  Hn 

max 12j112212 ,Ax  Axj;  , H,0 x,x 12  xjH n 

nn max 1 ,Ax, 1 1  1 C,x 1 0 C max  2 ,Axj, 2 2  2 C,x 2 0 C 

max 12  max  j

max 1   2 j  max  1   2 j

PERRON-FROBENIUS THEOREM FOR IRREDUCIBLE MATRICES ON QUATERNION DOUBLY STOCHASTIC MATRICES

Let A be an irreducible non-negative nn quaternion doubly stochastic matrix with period and spectral radius (A) r . Then the following statements hold

1. The number r is a positive real number and it is an eigen value of the quaternion doubly stochastic matrice , called the Perron-Frobenius eigen value. 2. The Perron-Frobenius eigen value r is simple, both right and left eigen spaces associated with r are one-dimensional. 3. has a left eigenvector V with eigen value r whose components are all positive. 4. Likewise, has a right eigenvector W with eigenvalue r whose componenets are all positive. 5. The only eigenvector whose components are all positive are those associated with the eigen value r.

6. The Perron-Frobenius eigen value satisfies the inequalities mini a ij r max i a ij jj DEFINITION (2) : A matrix is said to be a quaternion doubly stochastic matrix if

1. 0 aij 1 n 2. , a1ij  i 1,2,...,n j1 n 3.  a1ij  , j 1,2,...,n i1

1-NORM n n A max1 j n a i j the maximum absolute column sum A max1 j n a i j the maximum absolute row i1 i1 sum. INFINITY NORM

n A max1  i  n a i j j1

640 | IJREAMV04I1044021 DOI : 10.18231/2454-9150.2018.1419 © 2019, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-10, Jan 2019

INEQUALITIES FOR PERRON-FROBENIUS EIGEN VALUE:

For any non-negative quaternion doubly stochastic matrix A its Perron -Frobenius eigen value r satisfies the inequality .This is not specific to non-negative matrices for any matrix with an eigen value it is r maxi A ij  j true max A . i j ij

Any matrix induced norm satisfies the inequality A  for any eigen values  because, x is a corresponding eigenvector ,

A Ax / x   x / x   .

n The infinity norm of a matrix is the maximum of row sums A max A . Hence the desired inequality is  1 i m ij j1 exactly A  applied to the non-negative matrix A min A r . Given that is positive , then there exists a  i ij j

w positive eigenvector w such that Aw rw and the smallest components of w is 1. Then r (A )i the sum of the numbers in row of . Thus the minimum row sum gives a lower bound for r and this observation can be extended to all non- negative matrices.

THEOREM: Every eigen value of lies within atleast one of the GershgorinDisc D(aii ,¡ i ) .

PROOF:Let M(K)n be an eigen value of choose a corresponding eigenvector x (xj ) so that one component xi is equal to 1 and the others are of absolute values less than or equal to 1. x1i  and x1j  for ji . There always is such an x, which can be obtained simply by dividing any eigenvector by its component with largest modules.

So, splitting the sum Apply triangle inequality Ax x , aij x j  x i   aij x j a ii   j ji a x a x  a  ¡ for K,,H ¡¡ and AM(K) ij j ij j ii i    n . j i j i

III. CONCLUSION In this paper we can summarize that Perron-Fronbenius theory to general real and to a quaternion doubly stochastic matrices. We can discuss some properties and characterizations of quaternion doubly stochastic spectral radius on signature matrices; irreducible matrices and GershgorinDisc. Some norms and inequalities for Perron- Fronbenius eigen values are also discussed. REFERENCES [1] R. Brualdi, S. Parter and H. Schneider, “The diagonal equivalence of a nonnegative matrix” (to appear). [2] K.Gunasekaran and Seethadevi.R , “Characterization and Theorems on Quaternion doubly stochastic matrices”, JUSPS-A vol.29(4), 135-143 (2017) [3] M. Marcus and H. Minc, “A survey of matrix theory and matrix in equalities”, Allyn and Bacos, 1964. [4] M. Marcus and M. Newman, unpublished paper. [5] J. Maxfield and H. M ,”A doubly stochastic matrix equivalent to a given matrix” , Notices Amer. Math. Soc. 9 (1962), 309. [6] R. Sinkhorn, “A relationship between arbitrary positive matrices and doubly stochastic matrices” , Ann. Math. Statist. 35 (1964, 876-879.

641 | IJREAMV04I1044021 DOI : 10.18231/2454-9150.2018.1419 © 2019, IJREAM All Rights Reserved.