Advances in Linear Algebra & Matrix Theory, 2015, 5, 143-149 Published Online December 2015 in SciRes. http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2015.54014
Generating Totally Positive Toeplitz Matrix from an Upper Bidiagonal Matrix
Mohamed A. Ramadan1, Mahmoud M. Abu Murad2 1Department of Mathematics, Faculty of Science, Menoufia University, Al Minufya, Egypt 2Department of Mathematics, Faculty of Science, Zagazig University, Ash Sharqiyah, Egypt
Received 17 September 2015; accepted 29 November 2015; published 2 December 2015
Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract In this paper, we construct one of the forms of totally positive Toeplitz matrices from upper or lower bidiagonal totally nonnegative matrix. In addition, some properties related to this matrix involving its factorization are presented.
Keywords Totally Positive Matrix, Totally Nonnegative Matrix, Toeplitz Matrix, LU Factorization
1. Introduction Total positive matrices arise in many areas in mathematics, and there has been considerable interest lately in the study of these matrices. For background information see the most important survey in this field by T. Ando [1]. See also [2]. A matrix A is said to be totally positive, if every square submatrix has positive minors and A is said to be to- tally nonnegative, and if every square submatrix has nonnegative minors. While it is well known that many of the nontrivial examples of totally positive matrices are obtained by restricting certain kernels to appropriate fi- nite subsets of R (see, for example, Ando ([1], p. 212) or Pinkus ([3], p. 2). For Toeplitz matrices, that is, ma- n trices of the form Tt= − , a complete characterization of the total positivity, in terms of certain entire ( ij)ij,1= functions, has been studied in a series of references by Ando [1], Pinkus [3] and S.M. Fallat, C.R. Johnson [4]. Expressing a matrix as a product of lower triangle matrix L and an upper triangle matrix U is called a LU fac- torization. Such factorization is typically obtained by reducing a matrix to an upper triangular matrix from via row operation, that is, Gaussian elimination. The primary purpose of this paper is to provide a new totally positive matrix generated from a totally nonneg-
How to cite this paper: Ramadan, M.A. and Murad, M.M.A. (2015) Generating Totally Positive Toeplitz Matrix from an Up- per Bidiagonal Matrix. Advances in Linear Algebra & Matrix Theory, 5, 143-149. http://dx.doi.org/10.4236/alamt.2015.54014 M. A. Ramadan, M. M. A. Murad
ative one and to construct its factorization. The organization of our paper is as follows. In Section 2, we introduce our notation and give some auxiliary results which we use in the subsequent sections. In Section 3, we recall from [3] the Toeplitz matrices speci- n fied for the case tk = , on which our proofs heavily rely. In Section 4, we present the proofs of our main k results. In last section, we present the factorization of this resulted matrix.
2. Notation and Auxiliary Results 2.1. Notations In this subsection we introduce the notation that will be used in developing the paper. For kn, we denote by
Qkn, the set of all strictly increasing sequences of k integers chosen from {1, 2, , n} . For αα= { 1,, αk } , ββ={ 1,,, βk} ∈Q kn, we denote by Aαβ the kk× submatrix of A contained in the rows indexed by αα1,, k and columns indexed by ββ1,, k . A matrix A is called totally positive (abbreviated TP henceforth) and totally nonnegative (abbreviated TN) if detAαβ > 0 and detAαβ ≥ 0 , respectively, for all α, β ∈=Qkkn, , 1, 2, , n. If a totally nonnegative matrix is also nonsingular, we write NsTN. Definition 2.1.1 [3] A square lower (upper) triangular matrix A is called lower (upper) triangular positive matrix, denoted LTP
(UTP), if for all kn=1, , and for αα= { 1,, αββk} ,={ 1, ,, βk} ∈Q kn with the property that αβss≥ ( βαss≥ ) for sk=1, , , then detAαβ > 0 . Let I be the square identity matrix of order n, and for 1,≤≤ij n, we let Eij be the square standard basis ma- trix whose only nonzero entry is 1 that occurs in the (ij, ) position. A tridiagonal matrix that is also upper (lower) triangular is called an upper (lower) bidiagonal matrix. State- ments referring to just triangular or bidiagonal matrices without the adjectives “upper” or “lower” may be ap- plied to either case.
2.2. Auxiliary Results We use the following classic formula known as Cauchy-Binet formula and stated in the theorem below. Theorem 2.2.1 (Cauchy-Binet formula) ([4], p. 27). Let A be an nm× matrix and B be an mp× matrix then for each pair of indexed sets α ⊆ {1, , n} and β ⊆ {1, , p} of cardinality k, where 1≤≤k min{ nmp , , } , we have detAB[α , β] = ∑ detAB[αη ,]⋅ det[ ηβ , ] ηη, =k The following remarkable result is one of the most important and useful results in the study of TN matrices. This result first appeared in [5] see also [1] for another proof of this fact. n Theorem 2.2.2. Let Aa= be a square matrix of order n. Then A is NsTN if and only if A has an LU ( ij )ij,1= factorization, such that both L and U are NsTN square matrices. Using this theorem and Cauchy-Binet formula we have the following corollary. n Corollary 2.2.3 [6]. Let Aa= be a square matrix of order n. Then A is TP if and only if A has an LU ( ij )ij,1= factorization, such that both L and U are TP square matrices. We have the following theorem to prove both L and U are totally positive. n Theorem 2.2.4. Let Uu= be an upper triangular square matrix of order n satisfying ( ij )ij,1= U1, , kj + 1, , j + k > 0 for j=0,1, , nk − , kn=1, , n Then U is UTP (upper totally positive). Similarly, if Ll= is an lower triangular square matrix of order ( ij )ij,1= n satisfying Li +1, , i + k 1, , k > 0 for j=0,1, , nk − , kn=1, , . Then L is LTP (lower totally posi- tive).
144 M. A. Ramadan, M. M. A. Murad
In the sequel we will make use the the following lemma, see, e.g. [7]. Lemma 2.2.5 (Sylvester Identity) Partition square matrix T of order n, n > 2 , as:
cT1 12 c 2 = TTTT21 22 23 , cT3 32 c 4 where T22 square matrix of order n − 2 and ccc123,, and c4 are scalars. Define the submatrices
cT1 12 T12 c 2 TT21 22 TT22 23 TT1234= ,,,= TT= = TT21 22 TT22 23 cT3 32 T32 c 4
Then if T22 is non singular detTT det− det TT det detT = 14 23 detT22
3. Toeplitz Matrices
Assuming we are given a finite sequence t−nn, t −+1 ,,,,,, t − 101 tt tn of distinct real numbers, the associated To- n n eplitz matrix is defined by Tt= − or Tt= − . If we are given a one-sided finite sequence tt,, , t, ( ji)ij,1= ( ij)ij,1= 01 n then we understand this to mean that tk−k =0, = 1,2, , n in the above definition. Sequences that give rise to totally positive Toeplitz matrices have been totally characterized in terms of their generating functions, i.e. re- n k presentations of ∑ txk . kn=−
In our case, the normalization t0 =1 , the sequence tt01,, , tn gives rise to a totally positive Toeplitz matrix n n k Tt= − if and only if tx has the form ( ji)ij,1= ∑ k k=0
n ∏(1+ ixs ) λx s=1 e n ∏(1− jxs ) s=0 where λ ≥≥0,ijss 0, ≥ 0 . n Now consider the polynomial fx( ) =(1 + x) , the upper triangular Toeplitz matrix nnn n 01 2 n − 1 nn n 0 01 n − 2 n U = 00 0 n 1 n 0000 0 is TP.
145 M. A. Ramadan, M. M. A. Murad
4. Generating New Form of Toeplitz Matrix 4.1. Main Result Now we formalize the structure of our result by the following theorem.
Theorem 4.1.1. Assume that we are given the sequence xx11,, n− of (n −1) distinct positive real numbers. n Define the upper bidiagonal matrix Pp= by ( ij )ij,1=
xi if ji= + 1 pij = 0 otherwise
That is the sequence xx11,, n− lies on the superdiagonal. Then the matrix T defined as
nn nn r T k T= fP( ) = ∑∑ PP( ) rk=00 = rk is TP. Proof To prove this result we must note that
nn kkn n nn rrTTn n T===×= f( P) ∑∑PP( ) ∑P ∑ (P) UL rk=00 = rk r=0r k=0 k
n n k n r T where UP= ∑ is upper triangular matrix and LP= ∑( ) is lower triangular matrix. By corollary r=0 r k=0 2.2.3 A is TP if both U and L are TP. n n r So, want to prove UP= ∑ is upper TP. r=0 r
nn n n n−1 x1 xx 12 ∏ xi 01 2 n − 1 i=1 nn n n−2 0 xx2 ∏ i 01 n − 2 i=1 n U = 00 0 n xn−1 1 n 00 00 0 By Theorem 2.2.4 U is TP if nn n j j+11 jk +− nn n U1, , kj + 1, , j + k = q j−12 j jk+− >0 nn n jk−+1 jk − j
146 M. A. Ramadan, M. M. A. Murad
nn +1 22 = kk where qx∏∏( k) ( xnk− ) which is positive and kk=11 =
nn n j j+11 jk +− nn n j−12 j jk+− > 0 nn n jk−+1 jk − j
Since its submatrix of Toeplitz matrix. Illustrative Example Let we have the following sequence of distinct positive real numbers 1, 4, 3. Define the matrix A as:
0100 0040 A = 0003 0000
Then the matrix function
4444401234 fA( ) = A ++++ A A A A 01234
01234 44444T TT TT ×( AAAAA) ++++ ( ) ( ) ( ) ( ) 01234
2 34 =+(IAA46 +2 + 4 AA 34 +) ×+( IA 4 T + 6( A T) + 4( A T) +( A T) )
1000 0100 004 0 000120000 0100 0040 00012 000 0 0000 =+×4 +×64 +× + 0010 0003 000 0 000 0 0000 0001 0000 000 0 000 0 0000
1000 0000 0 0 00 0 000 0000 0100 1000 0 0 00 0 000 0000 × +×46 +× +×4 + 0010 0400 4 0 00 0 000 0000 0001 0030 01200 120000000
1 4 24 48 1 0 0 0 2897 3844 600 48 0 1 16 72 4 1 0 0 3844 5441 880 72 =×= 0 0 1 12 24 16 1 0 600 880 145 12 0 0 0 1 48 72 12 1 48 72 12 1 is TP.
147 M. A. Ramadan, M. M. A. Murad
4.2. Properties
nnk nnr T 1) Note that detT = 1 since TP=×=∑∑ (P) UL and rk=00rk =
detT= det UL = det U det L =×= 111
Using this property we prove the following lemma Lemma 4.2.1. The matrix T, as defined above has the following property
detT22= det TT 1 det 4 − det TT 2 det 3
where TTTT1234,,, and T22 are defined in Lemma 2.2.5. Proof The statement follows by Lemma 2.2.5 and the idea of detT = 1 . 2) Let P denote the square matrix of order n permutation matrix by the permutation
ppij= n−+ i1, n − j + 1 , 1,≤≤ij n, and suppose T is a square TP Toeplitz matrix. Then PTP= LU is TP too (see [7]). Moreover, ST−1 S is TP, where S is diagonal matrix with diagonal entries alternately 1 and -1. 3) The Hadamrd product of two TP toeplitz matrices is TP matrix too, that is if we are given two square TP = = = = ⋅ matrices Ttij and Wwij of order n. Then the Hadamard product Y yij tw ij ij is TP.
5. Factorization 5.1. Construct New Factorization Our aim is to write the new TP Toeplitz matrix T as a product of elementary matrices of a special form. For any ∈ r δ = kn{2, , } , we let Eek ( ) ij to be the elementary lower matrix whose entries are defined by
1 if ij= eij =δ if i = kj , = k − r 0 otherwise
r r Note that Ek (δ ) can be written as Ek(δδ) = IE + kk, − r, where I is square identity matrix of order n and Eij, th is square matrix of order n whose non-zero entry is a 1 in the (ij, ) position n. Also, notice that rr−1 EEkk(−=δδ) ( ) . r We use the elementary matrices Ek (δ ) to reduce Lower diagonal matrix to identity matrix. For example, we can consider the following 44× Lower diagonal matrix L 1000 4100 L = 6410 4641 It can be factorized as
121321 LEEEEEEI= 233444(464464) ( ) ( ) ( ) ( ) ( )
121321464464 = (EEEEEEI233444(111111)) ( ( )) ( ( )) ( ( )) ( ( )) ( ( ))
5.2. General Characterization We begin a definition and a result that characterize the TP Toeplitz matrix T in terms of the elementary matrices l Ek (1) . = Theorem 5.2.1. Any square Toeplitz matrix of oreder n, Ttij can be written as
148 M. A. Ramadan, M. M. A. Murad
1 uu12 13 u1n 100 0 01uu23 2n l21 10 0 T= UL = 00 1 u l l 10 3n 31 32 00 0 1lllnn123 n 1 T T TTTT u(nn−−12) u(n) n uu14 23 uu13 12 = E11 E 2 1 E 3121 11 E E 11 EI (( nn( )) ) (( ( )) ) (( 4332( )) ) (( ( )) ) (( ( )) ) (( ( )) ) l ll l ll 121321 31 32 41 2nn( −−21) 1 nn( ) ⋅(EEEE23(1111)) ( ( )) ( 34( )) ( ( )) ( En ( 1)) (En (1))
nk T nk ulnk−+1, nk −+ 2 ki,− 1 That is, T= Enk− +1111×× IEki−+ ∏∏(( nk−+2 ( )) ) ∏∏( k ( )) ki=22 = ki=22 = Illustrative Example Let 2897 3844 600 48 1 4 24 48 1 0 0 0 3844 5441 880 72 0 1 16 72 4 1 0 0 T = = × 600 880 145 12 0 0 1 12 24 16 1 0 48 72 12 1 0 0 0 1 48 72 12 1 The matrix in this example can be factorized as
12 T72 T48 T16 T24 TT4 12 3121 TE=( 44(111111)) ×( E( )) ×( E 4( )) ×( E 33( )) ×( E( )) ××( E 2( )) I ( ) ( ) ( ) ( ) ( ) ( ) 4 24 16 48 72 12 12 1 3 2 1 ××(EE23(11)) ( ( )) ×( E 3( 1)) ×( E 4( 1)) ×( E 4( 1)) ×( E 4( 1))
Note that the number of the factored matrices equal 2×( 12 +++++= 72 48 16 24 4) 352 matrix
References [1] Ando, T. (1987) Totally Positive Matrices. Linear Algebra and its Applications, 90, 165-219. http://dx.doi.org/10.1016/0024-3795(87)90313-2 [2] Fallat, S.M. (2007) Totally Positive and Totally Nonnegative Matrices. Chapter 21 in Handbook of Linear Algebra, In: Hogben, L., Ed., Chapman & Hall/CRC, Boca Raton. [3] Pinkus, A. (2010) Totally Positive Matrices, Cambridge Tracts. Vol.181, Cambridge Univ. Press, Cambridge. [4] Fallat, S.M. and Johnson, C.R. (2011) Totally Nonnegative Matrices, Princeton Ser. Appl. Math. Princeton University Press, Princeton, Oxford. [5] Cryer, C.W. (1973) The LU-Factorization of Totally Positive Matrices. Linear Algebra and its Applications, 7, 83-92. http://dx.doi.org/10.1016/0024-3795(73)90039-6 [6] Fallat, S.M. (2001) Bidiagonal Factorizations of Totally Nonnegative Matrices. American Mathematical Monthly, 108, 697-712. http://dx.doi.org/10.2307/2695613 [7] Adm, M. and Garloff, J. (2013) Interval of Totally Nonnegative Matrices. Linear Algebra and its Applications, 439, 3796-3806.
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