On the bounds for ℓp norms of Khatri-Rao and Tracy-Singh products of Cauchy-Toeplitz matrices

Selçuk J. Appl. Math. Selçuk Journal of Vol. 6. No. 2. pp. 43-52, 2005 Applied Mathematics

On the Bounds for`pNorms of Khatri-Rao and Tracy-Singh Products of Cauchy-Toeplitz Matrices

Hac¬Civciv1 and Ramazan Türkmen1

1_{Department of Mathematics, Art and Science Faculty of Selcuk University, Selçuklu}

42079, Konya, Turkey;

e-mail:hacicivciv@ mynet.com , rturkm en@ selcuk.edu.tr

Received: July 6, 2005

Summary. In this paper, we have established a lower bound and an upper bound for the `p norms of Khatri-Rao and Tracy-Singh products of Cauchy-Toeplitz matrices of the form Tn=[1= (g + (i j)h)]ni;j=1, where g and h 6= 0 are arbitrary numbers such that 0 < g=h < 1.

Key words: Cauchy-Toeplitz matrices, Kronecker product, Khatri-Rao prod-uct, Tracy-Singh prodprod-uct, Norm.

1. Introduction and Preliminaries

A Cauchy-Toeplitz matrix is a matrix which is both a Cauchy matrix ( i.e. (1=(xi yj))ni;j=1; xi6= yj) and a Toeplitz matrix ( i.e. (zi j)ni;j=1). In general Cauchy-Toeplitz matrix is being de…ned as

(1) Tn=

1 g + (i j)h

n

i;j=1

where g and h 6= 0 are arbitrary numbers and g=h is not an integer.

Recently, there have been several papers on the norms of Cauchy-Toeplitz ma-trix and Cauchy-Hankel mama-trix [6; 7; 8; 9]:In[9], a lower bound for the spectral norm of Cauchy-Toeplitz matrix was obtained by Tyrtyshnikov taking g = 1=2 and h = 1 in (1): Parter [8] proved that singular values could be related to eigen-values of certain Hermitian Toeplitz matrices corresponding to Laurent Fourier series.Turkmen and Bozkurt [2] have established bounds for the spectral norms of Cauchy-Toeplitz matrix in the forms (1) by taking g = 1=k and h = 1. Solak

and Bozkurt [3] obtained lower and upper bounds for the spectral norms of the Tn matrix that has given by (1). Liu[4] established a connection between the Khatri-Rao and Tracy-Singh products introduced by Khatri and Rao and Tracy and Singh, respectively, and present further results including matrix equalities and inequalities involving the two products and also gave two statistical ap-plications. Liu[5] obtained new inequalities involving Khatri-Rao products of positive semide…nite matrices. Additionally, we know that the Hadamard and Kronecker products play an important role in matrix methods for statistics, see e.g. [10 12], also these products are studied and applied widely in ma-trix theory and statistics; see, e.g.,[10; 11; 13 15]: For partitioned matrices,the Khatri-Rao product, viewed as a generalized Hadamard product, is discussed and used in [12; 16; 17 19] and the Tracy-Product, as a generalized Kronecker product, is discussed and applied in [20; 21]:

The purpose of this paper is to study the `pnorms of Khatri-Rao and Tracy-Singh products of two n n Cauchy-Toeplitz matrices in the form (1):In this section, we give some preliminaries. In section 2, we have studied the `p norms of Khatri-Rao product of two n n Cauchy-Toeplitz matrices is of the form (1) and obtained lower and upper bounds for these norms. In section 3, we have established lower and upper bounds for the `p norms of Tracy-Singh product of two n n Cauchy-Toeplitz matrices.

We now start with some preliminaries. Let A be any m n matrix. The `p norm of the matrix A is de…ned as

(2) _kAkp= m P i=1 n P j=1ja ijjp !1=p 1 _{p < 1:}

The Riemann Zeta function is de…ned as (s) = P1

n=1 1 ns

for complex values of s. while converging only for complex numbers s with Re s > 1, this function can be analytically continued to the whole complex plane ( with a single pole at s = 1 ).

The Hurwitz zeta function de…ned by (s; a) = P1

k=0 1 (k + a)s

for s 2 C; Re s > 1 and a 6= 0; 1; 2; ::: is a generalization of the Riemann Zeta function (s) = (s; 1):

Consider matrices A = (aij) and C = (cij) of order m n and B = (bkl) of order p q: Let A = (Aij) be partitioned with Aij of order mi njas the (i; j)th block

submatrix and let B = (Bkl) be partitioned with Bkl of order pk ql as the (k; l)th block submatrix (Pmi= m;Pnj= n;Ppk= p and Pql= q). Four matrix products of A and B, namely the Kronecker, Hadamard, Tracy-Singh and Khatri-Rao products, are de…ned as follows.

The Kronecker product, also known as tensor product or direct product, is de…ned to be

A B = (aijB);

where aij is the ijth scalar element of A = (aij); aijB is the ijth submatrix of order p q and A B is of order mp nq.

The Hadamard product, or the Schur product, is de…ned as A C = (aijcij);

where aij; cij and aijcij are the ijth scalar elements of A = (aij); C = (cij) and A C respectively, and A; C and A C are of order m n.

The Tracy-Singh product is de…ned to be

A B = (Aij B) with Aij B = (Aij Bkl)

where Aij is the ijth submatrix of order mi nj; Bkl is the klth submatrix of order pk ql; Aij Bkl is the klth submatrix of order mipk njql; Aij B is the ijth submatrix of order mip njq and A B is of order mp nq.

The Khatri-Rao product is de…ned as

A B = (Aij Bij)

where Aij is the ijth submatrix of order mi nj; Bij is the ijth submatrix of order pi qj; Aij Bij is the ijth submatrix of order mipi njqj and A B is of order (Pmipi) (Pnjqj) :

Throughout this paper we assume g and h 6= 0 are arbitrary numbers and 0 < g=h < 1.

2. `p norm of Khatri-Rao product of two n n Cauchy-Toeplitz ma-trices.

Theorem 1 Let the matrix Tn(n 2) be as in (1) for g and h 6= 0 arbitrary numbers and 0 < g=h < 1: If p is odd, then

kTn Tnkpp n 1 jhjp p; g h + p; g h + ( h g) p 2 + 1 (h)2p " p; g h + h g p 2 + p;g h h g p 2# + 1 (g)2p:

and kTn Tnkpp n 1 jhjp p; g h + 2( h g) p 2 + 1 (g h)2p + 1 (g + h)2p + 1 (g)2p and also if p is even, then

kTn Tnkpp n 1 jhjp p; g h + p; g h ( h g) p 2 + 1 (h)2p " p; g h h g p 2 + p;g h h g p 2# + 1 (g)2p and kTn Tnkpp n 1 jhjp p; g h 2 + 1 (g h)2p + 1 (g + h)2p + 1 (g)2p hold where k:kp(1 p < 1) is `pnorm and the operation “ ”is a Khatri-Rao product.

Proof. Let Tn(n 2) given in (1) be partitioned as

(3) Tn= T

(11) n Tn(12) Tn(21) Tn(22)

!

where Tn(ij) is the ijth submatrix of order mi nj with Tn(11) = Tn 1. Then Tn Tn , Khatri-Rao product of two Tn matrices, is obtained as

Tn Tn = T (11) n Tn(11) Tn(12) Tn(12) Tn(21) Tn(21) Tn(22) Tn(22) ! :

Using the `pnorm and Khatri-Rao de…nitions one may easily compute kTn Tnkp relative to the above Tn(ij) Tn(ij)

p as shown in (4) (4) _kTn Tnkpp= 2 P i;j=1 T_n(ij) T_n(ij) p p

T_n(11) T_n(11) p p = " n 1_P i;j=1 1 jg + (i j) hjp #2 (5) = n 2_P k=0 n k 1 jg khjp + n 2_P k=1 n k 1 jg + khjp 2 = (n 1) n 2_P k=0 1 jg khjp + n 2_P k=1 1 jg + khjp n 2_P k=0 k jg khjp + n 2_P k=1 k jg + khjp 2

We multiply equality (5) from the left and from the right by (n 1) 2_{to obtain}

(n 1) 2 T_n(11) T_n(11) p p = 1 (h)2p " n 2_P k=0 1 g h k p + n 2_P k=1 1 g h+ k p ! (6) 1 n 1 n 2_P k=0 k g h k p + n 2_P k=1 k g h+ k p !#2

Also we have following equalities,

(7) lim n!1 1 n 1 n 2_P k=0 k g h k p + n 2_P k=1 k g h+ k p ! = 0

and If p is odd then, (8) P1 k=0 1 k g_h p = 2 h g p + P1 k=0 1 k g_h p = 2 h g p + p; g h and, If p is even then,

(9) P1 k=0 1 k g_h p = 1 P k=0 1 k _hg p = p; g h :

Hence from (4) and from equalities (6); (7); (8); (9); if p is odd, then we have

kTn Tnkpp n 1 jhjp p; g h + p; g h + ( h g) p 2 + 1 (g)2p + 1 (h)2p 2 4 P1 k=1 1 k g_h p !2 + P1 k=1 1 k +g_h p !23 5

kTn Tnkpp n 1 jhjp p; g h + p; g h + ( h g) p 2 (10) + 1 (h)2p " p; g h + h g p 2 + p;g h h g p 2# + 1 (g)2p: On the other hand, if p is even, then we have

kTn Tnkpp n 1 jhjp p; g h + p; g h ( h g) p 2 (11) + 1 (h)2p " p; g h h g p 2 + p;g h h g p 2# + 1 (g)2p:

According to the cases of p (odd or even), these are upper bounds for kTn Tnkp_p.

For the lower bound, if we neglect the term _(h)1p

n 2_P k=1

1 jg

h+kj

p in the right - hand

side of the equality (6), if p is odd, then we have kTn Tnkpp n 1 hp 2 p; g h + 2( h g) p 2 + 1 (h)2p 2 4 2 1P k=1 1 k g_h p !2 + 2 1_P k=1 1 k + g_h p !23 5 + 1 (g)2p kTn Tnkpp n 1 jhjp p; g h + 2( h g) p 2 (12) + 1 (g h)2p + 1 (g + h)2p + 1 (g)2p: On the other hand, if p is even, then we have

(13) _kTn Tnkpp n 1 hp p; g h 2 + 1 (g h)2p + 1 (g + h)2p + 1 (g)2p from (4) and from equalities (6); (7); (8); (9):Thus, the proof of the theorem is completed from (10); (11); (12); (13):

3. `p norm of Tracy-Singh product of two n n Cauchy-Toeplitz matrices

Theorem 2Let the matrix Tn(n 2) be as in (1) for g and h 6= 0 are arbitrary numbers and 0 < g=h < 1: If p is odd, then

kTn Tnkpp n 1 jhjp p; g h + p; g h + ( h g) p ₊ 1 jgjp + 1 jhjp p; g h + p; g h 2 and kTn Tnkpp n 1 jhjp p; g h + 2( h g) p ₊ 1 (g h)p + 1 (g + h)p + 1 (g)p 2

and if p is even, then kTn Tnkp_p n 1 jhjp p; g h + p; g h ( h g) p ₊ 1 jgjp + 1 jhjp p; g h + p; g h 2( h g) p 2 kTn Tnkpp n 1 hp p; g h + 1 (g h)p + 1 (g + h)p + 1 (g)p 2

is valid where k:kp (1 p < 1) is `p norm and the operation “ ” Tracy-Singh product.

Proof. Let Tn(n 2) given in (1) be partitioned as (3): Then Tn Tn , Tracy-Singh product of two Tn matrices, is obtained as

Tn Tn= (Tn(ij) Tn)

where Tn(ij) Tn is the ijth submatrix of order min njn such that

T_n(ij) Tn=

Tn(ij) Tn(11) Tn(ij) Tn(12) Tn(ij) Tn(21) Tn(ij) Tn(22)

!

, i; j = 1; 2:

Using the `pnorm and Tracy-Singh de…nitions one may easily write

(14) _kTn Tnkp_p= 2 P i;j=1 2 P k;l=1 (ij; kl)

where (ij; kl) = Tn(ij) Tn(kl) p

pwith 1 i; j; k; l 2: Thus, from the `pnorm and Kronecker product de…nitions, we have

2 P k;l=1 (11; kl) = n 1_P i;j=1 1 jg + (i j) hjp ! " n 1_P i;j=1 1 jg + (i j) hjp ! (15) + n 1_P i=1 1 jg + (i n) hjp + 1 jgjp + n 1_P j=1 1 jg + (n j) hjp !# ; 2 P k;l=1 (12; kl) = n 1_P i;j=1 1 jg + (i n) hjp ! " n 1_P i;j=1 1 jg + (i j) hjp ! (16) + n 1_P i=1 1 jg + (i n) hjp + 1 jgjp + n 1_P j=1 1 jg + (n j) hjp !# ; 2 P k;l=1 (21; kl) = n 1_P i;j=1 1 jg + (n j) hjp ! " n 1_P i;j=1 1 jg + (i j) hjp ! (17) + n 1_P i=1 1 jg + (i n) hjp + 1 jgjp + n 1_P j=1 1 jg + (n j) hjp !# ; 2 P k;l=1 (22; kl) = 1 jgjp " n 1_P i;j=1 1 jg + (i j) hjp ! + 1 jgjp (3.5) + n 1_P j=1 1 jg + (n j) hjp ! + n 1_P i=1 1 jg + (i n) hjp : from the equalities (14) (18) we obtain

kTn Tnkpp = " n 1_P i;j=1 1 jg + (i j) hjp ! + n 1_P i=1 1 jg ihjp (18) + n 1_P j=1 1 jg + jhjp ! + 1 jgjp #2 :

We need to consider the following two cases.

Case 1. p is an odd integer. In this case, from (5); (7); (8) and (19) we have kTn Tnkp_p n 1 jhjp p; g h + p; g h + ( h g) p ₊ 1 jgjp + 1 jhjp 1 P i=1 1 i _hg p ! + P1 j=1 1 j +g_h p !!#2 kTn Tnkpp n 1 jhjp p; g h + p; g h + ( h g) p ₊ 1 jgjp (19) + 1 jhjp p; g h + p; g h 2 :

Case 2. p is an even integer. In this case, from (5); (7); (9) and (19) we have kTn Tnkp_p n 1 jhjp p; g h + p; g h ( h g) p ₊ 1 jgjp (20) + 1 jhjp p; g h + p; g h 2( h g) p 2 : Consequently, these are upper bounds for kTn Tnkpp.

Now if obtaining the equalities (12) and (13) is considered, in a similar way, if p is odd, then we have a lower bound for kTn Tnkp_p such that

kTn Tnkpp n 1 jhjp p; g h + 2( h g) p ₊ 1 (g h)p (21) + 1 (g + h)p + 1 (g)p 2 :

On the other hand if p is even, then we have a lower bound for kTn Tnkpp such that (22) _kTn Tnkpp n 1 jhjp p; g h + 1 (g h)p + 1 (g + h)p + 1 (g)p 2 : Thus, the proof of the theorem is completed from (20) (23):

References

1. Zhang F. (1999): Matrix Theory: Basic Results and Techniques, Springer-Verlag, New York.

2. Türkmen R. and Bozkurt D. (2002): On the bounds for the norms or Cauchy-Toeplitz and Cauchy-Hankel matrices, Applied Mathematics and Computation 132, 633-642.

3. Solak S. and Bozkurt D. (2003): On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices, Applied Mathematics and Computation 140, 231-238. 4. Liu S. (2002): Several inequalities involving Khatri-Rao products of positive senidef-inite matrices, Linear Algebra and its Applications 354, 175-186.

5. Liu S. (1999): Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra and its Applications 289, 267-277.

6. Bozkurt D. (1998): On the `p norms of Cauchy-Toeplitz matrices, Linear and

Multilinear Algebra, 44, 341-346.

7. Bozkurt D. (1996): On the bounds for the `p norm of almost Cauchy-Toeplitz

matrix, Turkish Journal of Mathematics 20(4), 544-552.

8. Parter S. V. (1986): On the disribution of the singular values of Toeplitz matrices, Linear Algebra and its Applications 80, 115-130.

9. Tyrtyshnikov E. E. (1991): Cauchy-Toeplitz matrices and some applications, Linear Algebra and its Applications 149, 1-18.

10. Styan G. P. H. (1973): Hadamard products and multivariate statistical analysis, Linear Algebra and its Applications 6, 217-240.

11. Magnus J. R. (1991): H. Neudecker, Matrix Di¤erential Calculus with Applications in Statistics and Econometrics, revised edition, Wiley, Chichester, UK.

12. Liu S. (1995): Contributions to matrix Calculus and Applications in Econometrics, Thesis Publishers, Amsterdam, The Netherlands.

13. Ando T. (1979): Concavity of certain maps on positive de…nite matrices and applications to Hadamard products, Linear Algebra Appl. 26, 203-241.

14. Horn R. A. (1990): The Hadamard product, Proc. Symp. Appl. Math. 40, 87-169.

15. Visick G. (1998): A uni…ed approach to the analysis of the Hadamard product of matrices using properties of the Kronecker product, Ph.D. Thesis, London Ubiversity, UK.

16. Khatri C. G., Rao C. R. (1968): Solutions to some functional equations and their applications to characterization of probability distributions, Sankhy¯a 30, 167-180. 17. Rao C. R. (1970): Estimation of heteroscedastic variances in linear models, J. Am. Statist. Assoc. 65, 161-172.

18. Rao C. R. and Kle¤e J. (1988): Estimation of Variance Components and Appli-cations, North-Holland, Amsterdam, The Netherlands.

19. Rao C. R., Rao M. B. (1998): Matrix Algebra and its Applications to Statistics and Econometrics, World Scienti…c, Singapore.

20. Koning R. H., Neudecker H., Wansbeek T. (1991): Block Kronecker product and vecb operator, Linear Algebra Appl. 149, 165-184.

21. Tracy D. S. and Singh R. P. (1972): A new matrix product and its applications in matrix di¤erentation, Statist. Neerlandica 26, 143-157.