^{1}

^{2}

^{*}

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.

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 [

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 ( [

trices of the form

functions, has been studied in a series of references by Ando [

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 [

fied for the case

results. In last section, we present the factorization of this resulted matrix.

In this subsection we introduce the notation that will be used in developing the paper. For

Definition 2.1.1 [

A square lower (upper) triangular matrix A is called lower (upper) triangular positive matrix, denoted LTP (UTP), if for all

Let I be the square identity matrix of order n, and for

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.

We use the following classic formula known as Cauchy-Binet formula and stated in the theorem below.

Theorem 2.2.1 (Cauchy-Binet formula) ( [

The following remarkable result is one of the most important and useful results in the study of TN matrices. This result first appeared in [

Theorem 2.2.2. Let

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 [

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

Then U is UTP (upper totally positive). Similarly, if

n satisfying

In the sequel we will make use the the following lemma, see, e.g. [

Lemma 2.2.5 (Sylvester Identity)

Partition square matrix T of order n,

where

Then if

Assuming we are given a finite sequence

eplitz matrix is defined by

then we understand this to mean that

presentations of

In our case, the normalization

where

Now consider the polynomial

is TP.

Now we formalize the structure of our result by the following theorem.

Theorem 4.1.1. Assume that we are given the sequence

Define the upper bidiagonal matrix

That is the sequence

is TP.

Proof

To prove this result we must note that

where

So, want to prove

By Theorem 2.2.4 U is TP if

where

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.

1) Note that

Using this property we prove the following lemma

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

where

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

[

3) The Hadamrd product of two TP toeplitz matrices is TP matrix too, that is if we are given two square TP

matrices

Our aim is to write the new TP Toeplitz matrix T as a product of elementary matrices of a special form. For any

Note that

We use the elementary matrices

For example, we can consider the following

It can be factorized as

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,

That is,

Illustrative Example

Let

The matrix in this example can be factorized as

Note that the number of the factored matrices equal

Mohamed A.Ramadan,Mahmoud M.Abu Murad, (2015) Generating Totally Positive Toeplitz Matrix from an Upper Bidiagonal Matrix. Advances in Linear Algebra & Matrix Theory,05,143-149. doi: 10.4236/alamt.2015.54014