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