# On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains

Authors: Jinwang Liu; Tao Wu; Dongmei Li; Jiancheng Guan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, zero prime factorizations for matrices over a unique factorization domain are studied. We prove that zero prime factorizations for a class of matrices exist. Also, we give an algorithm to directly compute zero left prime factorizations for this class of matrices.


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## Body

## 1. Introduction

Multidimensional linear systems theory has a wide range of applications in circuits, systems, control of networked systems, signal processing, and other areas (see, e.g., [1, 2]). Multivariate polynomial matrix theory is a well-established tool for these systems, since many problems in the analysis and synthesis of control systems can be well solved using multivariate polynomial matrix techniques [1–3].In recent years,n-D polynomial matrix factorizations have been widely studied [4–10]. In [11, 12], the zero left prime factorization problem was raised. This problem has been solved in [4–6]. The minor left prime factorization problem has been solved in [7, 10]. In the algorithms given in [7, 10], a fitting ideal of some module over the multivariate (n-D) polynomial ring needs to be computed. It is a little complicated.It is well known that a multivariate polynomial ring over a field is a unique factorization domain. Then, the following problem is interesting.Problem 1.
How to decide if a matrix with full row rank over a unique factorization domain has a zero left prime factorization?
In this paper, we will give a partial solution to this problem.


## 2. Preliminaries

LetR be a unique factorization domain. The set of all l×m matrices with entries from R is denoted by Rl×m. Let F∈Rl×ml<m. We denote the greatest common divisor of all l×l minors of F by dF. Let C∈Rl×l be a submatrix of F. By deleting C from F, we get a submatrix of F. This submatrix is denoted by F\C.LetC∈Rm×m. adjC denotes the adjoint matrix of C. acofijC denotes the i,jth algebraic cofactor of C.Definition 1.
LetF∈Rl×ml<m, and let C∈Rl×l be a submatrix of F. A minor of F consisting of l−1 columns from C and one column from F\C is said to be a related minor of C.
The following definition is from the multidimensional systems theory [13].Definition 2.
LetF∈Rl×m be of full row rank. Then, F is said to be zero left prime (ZLP) if the l×l minors of F generate the unit ideal R. Suppose F has a factorization F=CF1, where C∈Rl×landF1∈Rl×m. If F1 is ZLP, then this factorization is said to be a zero left prime factorization.


## 3. Main Results

First, we need a lemma.Lemma 1.
LetF=C,C¯∈Rl×ml<m, where C∈Rl×landC¯∈Rl×m−l. Then, the elements of adjC⋅C¯ are just all related minors of C (up to a sign).Proof.
LetC=cijl×l​andC¯=c¯ijl×m−l. Let adjC⋅C¯=bijl×m−l. Then,(1)bij=acof1iCc¯1j+⋯+acofliCc¯lj=detc11⋯c1i−1c¯1jc1i+1⋯c1l⋮⋮⋮⋮⋮cl1⋯cli−1c¯ljcli+1⋯cll,by Laplace Theorem. Thus, bij is a related minor of C (up to a sign). It is clear that they are just all related minors of C (up to a sign).
Now, we prove the main theorem of this paper.Theorem 1.
LetF∈Rl×ml<m. If there exists an l×l submatrix C of F such that detC is a common factor of all related minors of C, then there exists F1∈Rl×m such that F=CF1 and F1 is ZLP; i.e., F has a ZLP factorization.Proof.
We can change the order of the columns ofF such that the submatrix C consists of the left l columns of F. Thus, there exists an invertible matrix Q∈Rm×m such that FQ=C,C¯, where C∈Rl×landC¯∈Rl×m−l. Since detC is a common factor of all related minors of C, by Lemma 1, we have C−1C¯=adjC⋅C¯/detC∈Rl×m−l. Let(2)Q1=Il−C−1C¯0Im−l.
Then,Q1∈Rm×m. We have(3)FQQ1=C,C¯Q1=C,C¯Il−C−1C¯0Im−l=C,O.
Then,(4)F=C,OQ1−1Q−1by3=CIl,OQ1−1Q−1.
LetF1=Il,OQ1−1Q−1∈Rl×m. Then, F=CF1. Since F1 consists of the upper l rows of invertible matrix Q1−1Q−1, we have F1 is ZLP.Corollary 1.
LetF∈Rl×ml<m. If there exists an l×l submatrix C of F such that detC is a common factor of all related minors of C, then detC=dF.Proof.
Clearly,dFdetC. By Theorem 1, there exists F1∈Rl×m such that F=CF1. By Cauchy–Binet formula, we have detCdF. Therefore, detC=dF.Corollary 2.
LetF∈Rl×ml<m. If there exists an l×l submatrix C of F such that detC is a common factor of all related minors of C, then F is equivalent to C,O.Proof.
By Theorem1, there exists F1∈Rl×m such that F=CF1 and F1 is ZLP. By Quillen–Suslin theorem, there exists F2∈Rm−l×m such that F1T,F2TT is an invertible matrix. Since F=CF1=C,OF1T,F2TT, we have F being equivalent to C,O.
Now, letF∈Rl×ml<m. Suppose there exists an l×l submatrix C of F such that detC=dF. We can give an algorithm to directly compute the ZLP factorization of F.Algorithm 1.
(i)
Compute alll×l minors of F and dF.(ii)
Find anl×l submatrix C of F such that detC=dF.(iii)
Compute invertible matrixQ such that FQ=C,C¯.(iv)
LetQ1=Il−C−1C¯0Im−l and F1=Il,OQ1−1Q−1. Then, F=CF1.
Now, we give an example to illustrate this algorithm.Example 1.
LetR=ℤx,y, and let(5)F=6x2y+2xy2x2xy6x2y2+6x2y+2xy2+5xy2xy+2x2xy2+2xy+y.
Then,dF=2xy. Let(6)C=2x2xy2xy+2x2xy2+2xy+y.
Then,C is a 2×2 submatrix of F and detC=dF. Let(7)Q=001100010.
Then,FQ=C,C¯, where(8)C¯=6x2y+2xy6x2y2+6x2y+2xy2+5xy.
Thus,−C−1C¯=−y−3x. Let(9)Q1=10−y01−3x001.
Then,(10)Q1−1Q−1=y103x01100.
Let(11)F1=y103x01.
Then,F=CF1.



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*Source: 1684893-2020-04-22.xml*