Generating Totally Positive Toeplitz Matrix from an Upper Bidiagonal Matrix

Mohamed A. Ramadan¹, Mahmoud M. Abu Murad^2*

¹Department of Mathematics, Faculty of Science, Menoufia University, Al Minufya, Egypt.
²Department of Mathematics, Faculty of Science, Zagazig University, Ash Sharqiyah, Egypt.

DOI: 10.4236/alamt.2015.54014                

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

Received 17 September 2015; accepted 29 November 2015; published 2 December 2015

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 totally 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 finite subsets of R (see, for example, Ando ( [1] , p. 212) or Pinkus ( [3] , p. 2). For Toeplitz matrices, that is, ma-

trices of the form a complete characterization of the total positivity, in terms of certain entire

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 factorization. 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 nonnegative 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-

fied for the case, on which our proofs heavily rely. In Section 4, we present the proofs of our main

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 we denote by the set of all strictly increasing sequences of k integers chosen from. For, we denote by the submatrix of A contained in the rows indexed by and columns indexed by. A matrix A is called totally positive (abbreviated TP henceforth) and totally nonnegative (abbreviated TN) if and , respectively, for all. 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 and for with the property that () for, then.

Let I be the square identity matrix of order n, and for, we let be the square standard basis matrix whose only nonzero entry is 1 that occurs in the position.

A tridiagonal matrix that is also upper (lower) triangular is called an upper (lower) bidiagonal matrix. Statements referring to just triangular or bidiagonal matrices without the adjectives “upper” or “lower” may be applied 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 matrix and B be an matrix then for each pair of indexed sets and of cardinality k, where, we have

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.

Theorem 2.2.2. Let be a square matrix of order n. Then A is NsTN if and only if A has an LU

factorization, such that both L and U are NsTN square matrices.

Using this theorem and Cauchy-Binet formula we have the following corollary.

Corollary 2.2.3 [6] . Let be a square matrix of order n. Then A is TP if and only if A has an LU

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.

Theorem 2.2.4. Let be an upper triangular square matrix of order n satisfying

for,

Then U is UTP (upper totally positive). Similarly, if is an lower triangular square matrix of order

n satisfying for,. Then L is LTP (lower totally positive).

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, , as:

,

where square matrix of order and and are scalars. Define the submatrices

Then if is non singular

3. Toeplitz Matrices

Assuming we are given a finite sequence of distinct real numbers, the associated To-

eplitz matrix is defined by or. If we are given a one-sided finite sequence,

then we understand this to mean that 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-

presentations of.

In our case, the normalization, the sequence gives rise to a totally positive Toeplitz matrix

if and only if has the form

where.

Now consider the polynomial, the upper triangular Toeplitz matrix

is TP.

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 of distinct positive real numbers.

Define the upper bidiagonal matrix by

That is the sequence lies on the superdiagonal. Then the matrix T defined as

is TP.

Proof

To prove this result we must note that

where is upper triangular matrix and is lower triangular matrix. By corollary 2.2.3 A is TP if both U and L are TP.

So, want to prove is upper TP.

By Theorem 2.2.4 U is TP if

where which is positive and

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:

Then the matrix function

is TP.

4.2. Properties

1) Note that since and

Using this property we prove the following lemma

Lemma 4.2.1. The matrix T, as defined above has the following property

where and are defined in Lemma 2.2.5.

Proof

The statement follows by Lemma 2.2.5 and the idea of.

2) Let P denote the square matrix of order n permutation matrix by the permutation

, , and suppose T is a square TP Toeplitz matrix. Then is TP too (see

[7] ). Moreover, 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 and of order n. Then the Hadamard product 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, we let to be the elementary lower matrix whose entries are defined by

Note that can be written as, where I is square identity matrix of order n and is square matrix of order n whose non-zero entry is a 1 in the position n. Also, notice that.

We use the elementary matrices to reduce Lower diagonal matrix to identity matrix.

For example, we can consider the following Lower diagonal matrix L

It can be factorized as

5.2. General Characterization

We begin a definition and a result that characterize the TP Toeplitz matrix T in terms of the elementary matrices.

Theorem 5.2.1. Any square Toeplitz matrix of oreder n, can be written as

That is,

Illustrative Example

Let

The matrix in this example can be factorized as

Note that the number of the factored matrices equal

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[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
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