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Research Article

Characteristic min-polynomial of a triangular and diagonal strictly double R-astic matrices

[version 1; peer review: awaiting peer review]
PUBLISHED 08 Jul 2024
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Abstract

Background

Determinant and characteristic polynomials are important concepts related to square matrices. Due to the absence of additive inverse in max-plus algebra, the determinant of a matrix over max-plus algebra can be represented by a permanent. In addition, there are several types of square matrices over max-plus algebra, including triangular and diagonal strictly double ℝ -astic matrices. A special formula has been devised to determine the permanent and characteristic max-polynomial of those matrices. Another algebraic structure that is isomorphic with max-plus algebra is min-plus algebra.

Methods

Min-plus algebra is the algebraic structure of triple ( ℝ ε ′ , ⊕ ′ , ⊗ ) . Furthermore, square matrices over min plus algebra are defined by the set of matrices sized n × n , the entries of which are the elements of ℝ ε ′ . Because these two algebraic structures are isomorphic, the permanent and characteristic min-polynomial can also be determined for each square matrix over min-plus algebra, as well as the types of matrices.

Results

In this paper, we find out the special formulas for determining the permanent and characteristic min-polynomial of the triangular matrix and the diagonal strictly double ℝ -astic matrix.

Conclusions

We show that the formula for determining the characteristic min-polynomial of the two matrices is the same, for each triangular matrix and strictly double ℝ -astic matrix A , χ A ( x ) = ⨁ r = 0 , 1 , … , n ′ δ n − r ⊗ n r .

Keywords

min-plus algebra; permanent; characteristic min-polynomial; triangular matrix; diagonal strictly double R-astic matrix

1. Introduction

Let ε={}, with being a set of all real numbers. Max-plus algebra is defined as the set of ε that uses maximum () and addition () binary operations.1 The algebraic structure of triple (ε,,) is an idempotent semifield. In the application, max-plus algebra can be used to model and analyze a production system. Then, a square matrix in which its components are the element of ε is called matrix over max-plus algebra, denoted εn×n. In linear algebra, determinant and characteristic polynomials are important concepts related to square matrices. However, due to the absence of additive inverse, the determinant in max-plus algebra can be represented by permanent. Let Aεn×n, then its permanent is formulated by perm(A)=σPna1σ(1)a2σ(2)a(n), where Pn is the set of all permutations of the set {1,2,,n}.2 Then, the characteristic max-polynomial of A is formulated by χA(x)=perm(AxI), where I is identity matrix over max-plus algebra.1

Furthermore, there is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is defined as the set of ε={ε}, where being a set of all real numbers and ε=+ endowed with minimum () and addition () binary operations.3 Min-plus algebra can be used to determine the shortest path of a route. Similarly, consider the set of matrices sized n×n, the entries of which are the elements of ε. This set is known as the set of square matrices over min-plus algebra. The permanent and characteristic min-polynomials in min-plus algebra have also been determined by Siswanto et al.4 and Watanabe et al.5

In addition to discussing square matrices in general, some researchers also discuss special matrices. Research from Jafari and Hosseinyazdi6 has found the special formulas for determining characteristic max-polynomials of triangular matrices and diagonal strictly double -astic matrices. According to Jafari and Hosseinyazdi’s research and the min-plus algebraic structure, this research will discuss the characteristic min-polynomials of triangular matrices and diagonal strictly double -astic matrices.

2. Methods

This research method was a literature review by using references to linear algebra, max-plus algebra, and min-plus algebra in books, journals, or articles. The aim of this review is to expand the research on min-plus algebra based on research in linear algebra and max-plus algebra, and also determine the connections between them.

The research begins by defining the triangular matrix and strictly double -astic matrix over min-plus algebra according to Jafari and Hosseinyazdi’s research on max-plus algebra. Next, we try to find the special formula for determining the permanent of those matrices. After we find its special formula, we will simplify the way to determine the characteristic min-polynomial of normal square matrices over min-plus algebra for triangular matrix and strictly double -astic matrix.

Also in this section, we give a short introduction to the min-plus algebra and square matrix over min-plus algebra. The min-plus algebra is defined as the set of ε={ε}, where being a set of all real numbers and ε=+ endowed with minimum () and addition () binary operations so that every a,bε applies

ab=min(a,b)andab=a+b

Following3 for a,b,c,x,yε, both operations and are associative, we have (ab)c=a(bc) and (ab)c=a(bc). These operations are also commutative, we have ab=b'a and ab=ba. We have the identity element ε=+ with respect to such that aε=a=εa. Then we have the identity element e=0 with respect to such that ae=a=ea. And if xε, there exist the unique inverse y=x of x with respect to , we have xy=e. But for the operation, there is no unique inverse. However, the minimum operation is idempotent, we have aa=a. Based on this description, the algebraic structure of min-plus algebra is idempotent semifield.

Definition 2.1:

For xε and k, the k-power of x is defined by

xk=xx.ktimes

We defined square matrix over min-plus algebra, denoted by εn×n, as the set of all n×n matrices with entries in ε for positive integers n. We define the several operations in εn×n according to.7

  • (1) If A and Bεn×n then (AB)ij=min{aij,bij}.

  • (2) If A and Bεn×n then (AB)ij=  k=1 n(aikbkj).

  • (3) If Aεn×n and αε then we have (αA)ij=αaij.

  • (4) We have an identity matrix over min-plus algebra with respect to , where

    I=(0εε0),

such that AI=A=IA.

  • (5) Given Aεn×n and k, the k-power of A is defined by

    Ak=AA.ktimes

The following is the definition of the formula for determining the permanent and characteristic min-polynomials according to.4,5

Definition 2.2:

For a matrix Aεn×n the permanent of A is defined as

perm(A)=σPn'i=1n(a(i))
with σ and Pn is a set of all permutations from {1,2,,n}.

Definition 2.3:

Let Aεn×n. Then the characteristic min-polynomial of A is defined to be

χ(x)A=perm(IxA).

3. Characteristic min-polynomial of a triangular matrix

Definition 3.1:

Let Aεn×n. The square matrix A is called triangular matrix if all entries of above or below diagonal are equal to ε.

Theorem 3.1:

If Aεn×n then the permanent of the triangular matrix A is

perm(A)=a11a22a33ann.

Proof:

The theorem will be proved for the upper triangular matrix using induction.

  • (1) For n=2, we have

    A=(a11a12εa22).

So, it has been proven true that

perm(A)=perm(a11a12εa22)=(a11a22)(a12ε)=(a11a22)ε=a11a22
  • (2) Suppose this is true for an upper triangular matrix A sized (n1)×(n1). Then, consider A as an n×n matrix. The permanent matrix A can be calculated by expanding perm(A) along the first column. The purpose of this expansion is to eliminate the elements in the first column one by one, along with the removal of rows and columns according to the location of the elements. We have

    perm(A)=(a11a12εa22a1na2nεεann)=a11perm(a22a23εa33a2na3nεεann)εperm(a12a13εa33a1na3nεεann)εperm(a12a13a1na22a23a2nεa(n1)(n1)a(n1)n)=(a11a22ann)εε=a11a22ann.

Theorem 3.2:

If Aεn×n then the characteristic min-polynomial of the triangular matrix A is

χA(x)=  r=0' nδnrnr
where δ0=0, and δnr, for r=0,1,,n1 is the sum of the nr smallest diagonal elements.

Proof:

The theorem will be proved for the upper triangular matrix. First, we recall the formula for determining the characteristic min-polynomial of a square matrix. Then, apply the formula to the triangular matrix. We have

χA(x)=perm(IxA)=perm(xa11a1nεxann).

The matrix IxA is also the upper triangular matrix. So, based on Theorem 1, we obtained

(1)
χA(x)=(xa11)(xa22)(xann).

Furthermore, equation (1) can be written as

χA(x)=xn(a11a22ann)xn1[(a11a22)(a11ann)(a22a33)(a22ann)(a(n1)(n1)ann)]xn2[(a11a22a33)(a11a22ann)(a22a33a44)(a22a33ann)(a(n2)(n2)a(n1)(n1)ann)]xn3(a11a22ann)
or it can be simplified to
(2)
χA(x)=xn  r=0' naiixn1ijC2N'(aiiajj)xn2ijkC3N'(aiiajjakk)xn3(a11ann)
where N={1,2,,n}. The equation (2) also can be written as
(3)
χA(x)=xnδ1xn1δ2xn2δ3xn3δn
where δ1=r=0,,naii, δ2=ijC2N(aiiajj), δ3=ijkC3N(aiiajjakk), and δn=(a11ann). In addition, it can be known that δ1 is the smallest diagonal element, δ2 is the result of the sum of 2 smallest diagonal elements, δ3 is the result of the sum of 3 smallest diagonal elements, and δn is the result of the sum of all diagonal elements. If the index n in δ is converted to nr, the characteristic min-polynomial of the triangular matrix in (3) can be simplified to
χA(x)=  r=0' nδnrnr.

4. Characteristic min-polynomial of a diagonal strictly double -astic matrix

Definition 4.1:

Let Aεn×n. A square matrix A is diagonal strictly double -astic, if it satisfies

  • (1) aij+,

  • (2) on each row and on each column of A, there is only one finite element, and

  • (3) the finite element on (2) is the diagonal element on the matrix A.

Theorem 4.1:

If Aεn×n then the permanent of the diagonal strictly double -astic matrix A is

perm(A)=a11a22a33ann.

Proof:

The theorem will be proved using induction.

  • (1) For n=2, we have

    A=(a11εεa22).

So, it has been proven true that

perm(A)=perm(a11εεa22)=(a11a22)(εε)=(a11a22)ε=a11a22
  • (2) Suppose this is true for a diagonal strictly double -astic matrix A sized (n1)×(n1). Then, consider A as an n×n matrix. The permanent matrix A can be calculated by expanding perm(A) along the first column. The purpose of this expansion is to eliminate the elements in the first column one by one, along with the removal of rows and columns according to the location of the elements. We have

    perm(A)=(a11εεa22εεεεann)=a11perm(a22εεa33εεεεann)εperm(εεεa33εεεεann)εperm(εεa22εεεεεa(n1)(n1))=(a11a22ann)εε=a11a22ann.

Theorem 4.2:

If Aεn×n then the characteristic min-polynomial of the diagonal strictly double -astic matrix A is

χA(x)=  r=0' nδnrnr
where δ0=0, and δnr, for r=0,1,,n1 is the sum of the nr smallest diagonal elements.

Proof:

First, we recall the formula for determining the characteristic min-polynomial of a square matrix. Then, apply the formula to the diagonal strictly double -astic matrix. We have

χA(x)=perm(IxA)=perm(xa11εεxann).

The matrix IxA is also the diagonal strictly double -astic matrix. So, based on Theorem 3, we obtained

(4)
χA(x)=(xa11)(xa22)(xann).

Furthermore, equation (4) can be written as

χA(x)=xn(a11a22ann)xn1[(a11a22)(a11ann)(a22a33)(a22ann)(a(n1)(n1)ann)]xn2[(a11a22a33)(a11a22ann)(a22a33a44)(a22a33ann)(a(n2)(n2)a(n1)(n1)ann)]xn3(a11a22ann)
or it can be simplified to
(5)
χA(x)=xn  r=0' naiixn1ijC2N'(aiiajj)xn2ijkC3N'(aiiajjakk)xn3(a11ann)
where N={1,2,,n}. The equation (5) also can be written as
(6)
χA(x)=xnδ1xn1δ2xn2δ3xn3δn
where δ1=r=0,,naii, δ2=ijC2N(aiiajj), δ3=ijkC3N(aiiajjakk), and δn=(a11a22ann). In addition, it can be known that δ1 is the smallest diagonal element, δ2 is the result of the sum of 2 smallest diagonal elements, δ3 is the result of the sum of 3 smallest diagonal elements, and δn is the result of the sum of all diagonal elements. If the index n in δ is converted to nr, the characteristic min-polynomial of the diagonal strictly double -astic matrix in (6) can be simplified to
χA(x)=  r=0' nδnrnr.

5. Conclusions

It can be concluded that the permanent of a triangular matrix and a diagonal strictly double -astic matrix is the same, which is the product (concerning the operation) of its diagonal entries. Furthermore, the formula for determining the characteristic min-polynomial of these matrices is the same too. This formula is more concise than the formula for determining the characteristic min-polynomial of normal square matrix.

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Siswanto S and Al Maghribi SM. Characteristic min-polynomial of a triangular and diagonal strictly double R-astic matrices [version 1; peer review: awaiting peer review]. F1000Research 2024, 13:771 (https://doi.org/10.12688/f1000research.147931.1)
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VERSION 1 PUBLISHED 08 Jul 2024
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Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions
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