F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.147931.1 Research Article Articles Characteristic min-polynomial of a triangular and diagonal strictly double R-astic matrices

[version 1; peer review: awaiting peer review]

 Siswanto Siswanto Conceptualization Formal Analysis Funding Acquisition Methodology Project Administration Supervision Writing – Review & Editing https://orcid.org/0000-0001-8168-734X a 1 Al Maghribi Sahmura Maula Investigation Validation Visualization Writing – Original Draft Preparation 1 Mathematics, Universitas Sebelas Maret, Surakarta, Central Java, 57126, Indonesia sis.mipa@staff.uns.ac.id

No competing interests were disclosed.

 8 7 2024 2024 13 771 6 5 2024 Copyright: © 2024 Siswanto S and Al Maghribi SM 2024 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 '   n δ n − r ⊗ n r .

 min-plus algebra; permanent; characteristic min-polynomial; triangular matrix; diagonal strictly double R-astic matrix Universitas Sebelas Maret 288/UN27.22/PT.01.03/2023 Universitas Sebelas Maret through a Fundamental Research Grant scheme competition in 2024 with assignment agreement letter number 194.2/UN27.22/PT.01.03/2024 The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. 1. Introduction

Let ℝ ε = ℝ ∪ { − ∞ } , with ℝ being a set of all real numbers. Max-plus algebra is defined as the set of ℝ ε that uses maximum ( ⊕ ) and addition ( ⊗ ) binary operations. 1 The algebraic structure of triple ( ℝ 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 ℝ ε is called matrix over max-plus algebra, denoted ℝ ε n × 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 A ∈ ℝ ε n × n , then its permanent is formulated by perm ( A ) = ⨁ σ ∈ P n a 1 σ ( 1 ) ⊗ a 2 σ ( 2 ) ⊗ … ⊗ a nσ ( n ) , where P n is the set of all permutations of the set { 1 , 2 , … , n } . 2 Then, the characteristic max-polynomial of A is formulated by χ A ( x ) = perm ( A ⊕ x ⊗ I ) , 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 ℝ ε ′ = ℝ ∪ { ε ′ } , where ℝ 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 ℝ 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 Hosseinyazdi 6 has found the special formulas for determining characteristic max-polynomials of triangular matrices and diagonal strictly double ℝ -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 ℝ -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 ℝ -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 ℝ -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 ℝ ε ′ = ℝ ∪ { ε ′ } , where ℝ being a set of all real numbers and ε ′ = + ∞ endowed with minimum ( ⊕ ′ ) and addition ( ⊗ ) binary operations so that every a , b ∈ ℝ ε ′ applies a ⊕ ′ b = min ( a , b ) and a ⊗ b = a + b

Following 3 for a , b , c , x , y ∈ ℝ ε ′ , both operations ⊕ ′ and ⊗ are associative, we have ( a ⊕ ′ b ) ⊕ ′ c = a ⊕ ′ ( b ⊕ ′ c ) and ( a ⊗ b ) ⊗ c = a ⊗ ( b ⊗ c ) . These operations are also commutative, we have a ⊕ ′ b = b ⊕ ' a and a ⊗ b = b ⊗ a . We have the identity element ε ′ = + ∞ with respect to ⊕ ′ such that a ⊕ ′ ε ′ = a = ε ′ ⊕ ′ a . Then we have the identity element e = 0 with respect to ⊗ such that a ⊗ e = a = e ⊗ a . And if x ≠ ε ′ , there exist the unique inverse y = − x of x with respect to ⊗ , we have x ⊗ y = e . But for the ⊕ ′ operation, there is no unique inverse. However, the minimum operation is idempotent, we have a ⊕ ′ a = a . Based on this description, the algebraic structure of min-plus algebra is idempotent semifield.

For x ∈ ℝ ε ′ and k ∈ ℕ , the k -power of x is defined by x ⊗ k = x ⊗ … ⊗ x . ⏟ k times

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

If A and B ∈ ℝ ε ′ n × n then ( A ⊕ ′ B ) ij = min { a ij , b ij } .

If A and B ∈ ℝ ε ′ n × n then ( A ⊗ B ) ij = ⨁     k = 1 ′   n ( a ik ⊗ b kj ) .

If A ∈ ℝ ε ′ n × n and α ∈ ℝ ε ′ then we have ( α ⊗ A ) ij = α ⊗ a ij .

We have an identity matrix over min-plus algebra with respect to ⊗ , where I = ( 0 ⋯ ε ′ ⋮ ⋱ ⋮ ε ′ ⋯ 0 ) ,

such that A ⊗ I = A = I ⊗ A .

Given A ∈ ℝ ε ′ n × n and k ∈ ℕ , the k -power of A is defined by A ⊗ k = A ⊗ … ⊗ A . ⏟ k times

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

For a matrix A ∈ ℝ ε ′ n × n the permanent of A is defined as perm ( A ) = ⨁ σ ∈ P n ' ⨂ i = 1 n ( a iσ ( i ) ) with σ and P n is a set of all permutations from { 1 , 2 , … , n } .

Let A ∈ ℝ ε ′ n × n . Then the characteristic min-polynomial of A is defined to be χ = perm ( I ⊗ x ⊕ ′ A ) .

3. Characteristic min-polynomial of a triangular matrix

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

If A ∈ ℝ ε ′ n × n then the permanent of the triangular matrix A is perm ( A ) = a 11 ⊗ a 22 ⊗ a 33 ⊗ … ⊗ a nn .

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

For n = 2 , we have A = ( a 11 a 12 ε ′ a 22 ) .

So, it has been proven true that perm ( A ) = perm ( a 11 a 12 ε ′ a 22 ) = ( a 11 ⊗ a 22 ) ⊕ ′ ( a 12 ⊗ ε ′ ) = ( a 11 ⊗ a 22 ) ⊕ ′ ε ′ = a 11 ⊗ a 22

Suppose this is true for an upper triangular matrix A sized ( n − 1 ) × ( n − 1 ) . 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 ) = ( a 11 a 12 ε ′ a 22 … a 1 n … a 2 n ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a nn ) = a 11 ⊗ perm ( a 22 a 23 ε ′ a 33 … a 2 n … a 3 n ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a nn ) ⊕ ′ ε ′ ⊗ perm ( a 12 a 13 ε ′ a 33 … a 1 n … a 3 n ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a nn ) ⊕ ′ … ⊕ ′ ε ′ ⊗ perm ( a 12 a 13 … a 1 n a 22 a 23 … a 2 n ⋮ ⋮ ⋱ ⋮ ε ′ ⋯ a ( n − 1 ) ( n − 1 ) a ( n − 1 ) n ) = ( a 11 ⊗ a 22 ⊗ … ⊗ a nn ) ⊕ ′ ε ′ ⊕ ′ … ⊕ ′ ε ′ = a 11 ⊗ a 22 ⊗ … ⊗ a nn .

∎

If A ∈ ℝ ε ′ n × n then the characteristic min-polynomial of the triangular matrix A is χ A ( x ) = ⨁     r = 0 '   n δ n − r ⊗ n r where δ 0 = 0 , and δ n − r , for r = 0 , 1 , … , n − 1 is the sum of the n − r smallest diagonal elements.

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 χ A ( x ) = perm ( I ⊗ x ⊕ ′ A ) = perm ( x ⊕ ′ a 11 ⋯ a 1 n ⋮ ⋱ ⋮ ε ′ ⋯ x ⊕ ′ a nn ) .

The matrix I ⊗ x ⊕ ′ A is also the upper triangular matrix. So, based on Theorem 1, we obtained χ A ( x ) = ( x ⊕ ′ a 11 ) ⊗ ( x ⊕ ′ a 22 ) ⊗ … ⊗ ( x ⊕ ′ a nn ) .

Furthermore, equation (1) can be written as χ A ( x ) = x n ⊕ ′ ( a 11 ⊕ ′ a 22 ⊕ ′ … ⊕ ′ a nn ) ⊗ x n − 1 ⊕ ′ [ ( a 11 ⊗ a 22 ) ⊕ ′ … ⊕ ′ ( a 11 ⊗ a nn ) ⊕ ′ ( a 22 ⊗ a 33 ) ⊕ ′ … ⊕ ′ ( a 22 ⊗ a nn ) ⊕ ′ … ⊕ ′ ( a ( n − 1 ) ( n − 1 ) ⊗ a nn ) ] ⊗ x n − 2 ⊕ ′ [ ( a 11 ⊗ a 22 ⊗ a 33 ) ⊕ ′ … ⊕ ′ ( a 11 ⊗ a 22 ⊗ a nn ) ⊕ ′ ( a 22 ⊗ a 33 ⊗ a 44 ) ⊕ ′ … ⊕ ′ ( a 22 ⊗ a 33 ⊗ a nn ) ⊕ ′ … ⊕ ′ ( a ( n − 2 ) ( n − 2 ) ⊗ a ( n − 1 ) ( n − 1 ) ⊗ a nn ) ] ⊗ x n − 3 ⊕ ′ … ⊕ ′ ( a 11 ⊗ a 22 ⊗ … ⊗ a nn ) or it can be simplified to χ A ( x ) = x n ⊕ ′ ⨁     r = 0 '   n a ii ⊗ x n − 1 ⊕ ′ ⨁ ij ∈ C 2 N ' ( a ii ⊗ a jj ) ⊗ x n − 2 ⊕ ′ ⨁ ijk ∈ C 3 N ' ( a ii ⊗ a jj ⊗ a kk ) ⊗ x n − 3 ⊕ ′ … ⊕ ′ ( a 11 ⊗ … ⊗ a nn ) where N = { 1 , 2 , … , n } . The equation (2) also can be written as χ A ( x ) = x n ⊕ ′ δ 1 ⊗ x n − 1 ⊕ ′ δ 2 ⊗ x n − 2 ⊕ ′ δ 3 ⊗ x n − 3 ⊕ ′ … ⊕ ′ δ n where δ 1 = ⨁ r = 0 , … , n ′ a ii , δ 2 = ⨁ ij ∈ C 2 N ′ ( a ii ⊗ a jj ) , δ 3 = ⨁ ijk ∈ C 3 N ′ ( a ii ⊗ a jj ⊗ a kk ) , and δ n = ( a 11 ⊗ … ⊗ a nn ) . In addition, it can be known that δ 1 is the smallest diagonal element, δ 2 is the result of the sum of 2 smallest diagonal elements, δ 3 is the result of the sum of 3 smallest diagonal elements, and δ n is the result of the sum of all diagonal elements. If the index n in δ is converted to n − r , the characteristic min-polynomial of the triangular matrix in (3) can be simplified to χ A ( x ) = ⨁     r = 0 '   n δ n − r ⊗ n r .

∎

4. Characteristic min-polynomial of a diagonal strictly double <inline-formula> <mml:math display="inline"> <mml:mi>ℝ</mml:mi> </mml:math> </inline-formula> <italic toggle="yes">-astic</italic> matrix

Let A ∈ ℝ ε ′ n × n . A square matrix A is diagonal strictly double ℝ - astic, if it satisfies

a ij ≤ + ∞ ,

on each row and on each column of A , there is only one finite element, and

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

If A ∈ ℝ ε ′ n × n then the permanent of the diagonal strictly double ℝ - astic matrix A is perm ( A ) = a 11 ⊗ a 22 ⊗ a 33 ⊗ … ⊗ a nn .

The theorem will be proved using induction.

For n = 2 , we have A = ( a 11 ε ′ ε ′ a 22 ) .

So, it has been proven true that perm ( A ) = perm ( a 11 ε ′ ε ′ a 22 ) = ( a 11 ⊗ a 22 ) ⊕ ′ ( ε ′ ⊗ ε ′ ) = ( a 11 ⊗ a 22 ) ⊕ ′ ε ′ = a 11 ⊗ a 22

Suppose this is true for a diagonal strictly double ℝ - astic matrix A sized ( n − 1 ) × ( n − 1 ) . 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 ) = ( a 11 ε ′ ε ′ a 22 … ε ′ … ε ′ ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a nn ) = a 11 ⊗ perm ( a 22 ε ′ ε ′ a 33 … ε ′ … ε ′ ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a nn ) ⊕ ′ ε ′ ⊗ perm ( ε ′ ε ′ ε ′ a 33 … ε ′ … ε ′ ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a nn ) ⊕ ′ … ⊕ ′ ε ′ ⊗ perm ( ε ′ ε ′ a 22 ε ′ … ε ′ … ε ′ ⋮ ⋮ ε ′ ε ′ ⋱ ⋮ ⋯ a ( n − 1 ) ( n − 1 ) ) = ( a 11 ⊗ a 22 ⊗ … ⊗ a nn ) ⊕ ′ ε ′ ⊕ ′ … ⊕ ′ ε ′ = a 11 ⊗ a 22 ⊗ … ⊗ a nn .

∎

If A ∈ ℝ ε ′ n × n then the characteristic min-polynomial of the diagonal strictly double ℝ - astic matrix A is χ A ( x ) = ⨁     r = 0 '   n δ n − r ⊗ n r where δ 0 = 0 , and δ n − r , for r = 0 , 1 , … , n − 1 is the sum of the n − r smallest diagonal elements.

First, we recall the formula for determining the characteristic min-polynomial of a square matrix. Then, apply the formula to the diagonal strictly double ℝ - astic matrix. We have χ A ( x ) = perm ( I ⊗ x ⊕ ′ A ) = perm ( x ⊕ ′ a 11 ⋯ ε ′ ⋮ ⋱ ⋮ ε ′ ⋯ x ⊕ ′ a nn ) .

The matrix I ⊗ x ⊕ ′ A is also the diagonal strictly double ℝ - astic matrix. So, based on Theorem 3, we obtained χ A ( x ) = ( x ⊕ ′ a 11 ) ⊗ ( x ⊕ ′ a 22 ) ⊗ … ⊗ ( x ⊕ ′ a nn ) .

Furthermore, equation (4) can be written as χ A ( x ) = x n ⊕ ′ ( a 11 ⊕ ′ a 22 ⊕ ′ … ⊕ ′ a nn ) ⊗ x n − 1 ⊕ ′ [ ( a 11 ⊗ a 22 ) ⊕ ′ … ⊕ ′ ( a 11 ⊗ a nn ) ⊕ ′ ( a 22 ⊗ a 33 ) ⊕ ′ … ⊕ ′ ( a 22 ⊗ a nn ) ⊕ ′ … ⊕ ′ ( a ( n − 1 ) ( n − 1 ) ⊗ a nn ) ] ⊗ x n − 2 ⊕ ′ [ ( a 11 ⊗ a 22 ⊗ a 33 ) ⊕ ′ … ⊕ ′ ( a 11 ⊗ a 22 ⊗ a nn ) ⊕ ′ ( a 22 ⊗ a 33 ⊗ a 44 ) ⊕ ′ … ⊕ ′ ( a 22 ⊗ a 33 ⊗ a nn ) ⊕ ′ … ⊕ ′ ( a ( n − 2 ) ( n − 2 ) ⊗ a ( n − 1 ) ( n − 1 ) ⊗ a nn ) ] ⊗ x n − 3 ⊕ ′ … ⊕ ′ ( a 11 ⊗ a 22 ⊗ … ⊗ a nn ) or it can be simplified to χ A ( x ) = x n ⊕ ′ ⨁     r = 0 '   n a ii ⊗ x n − 1 ⊕ ′ ⨁ ij ∈ C 2 N ' ( a ii ⊗ a jj ) ⊗ x n − 2 ⊕ ′ ⨁ ijk ∈ C 3 N ' ( a ii ⊗ a jj ⊗ a kk ) ⊗ x n − 3 ⊕ ′ … ⊕ ′ ( a 11 ⊗ … ⊗ a nn ) where N = { 1 , 2 , … , n } . The equation (5) also can be written as χ A ( x ) = x n ⊕ ′ δ 1 ⊗ x n − 1 ⊕ ′ δ 2 ⊗ x n − 2 ⊕ ′ δ 3 ⊗ x n − 3 ⊕ ′ … ⊕ ′ δ n where δ 1 = ⨁ r = 0 , … , n ′ a ii , δ 2 = ⨁ ij ∈ C 2 N ′ ( a ii ⊗ a jj ) , δ 3 = ⨁ ijk ∈ C 3 N ′ ( a ii ⊗ a jj ⊗ a kk ) , and δ n = ( a 11 ⊗ a 22 ⊗ … ⊗ a nn ) . In addition, it can be known that δ 1 is the smallest diagonal element, δ 2 is the result of the sum of 2 smallest diagonal elements, δ 3 is the result of the sum of 3 smallest diagonal elements, and δ n is the result of the sum of all diagonal elements. If the index n in δ is converted to n − r , the characteristic min-polynomial of the diagonal strictly double ℝ - astic matrix in (6) can be simplified to χ A ( x ) = ⨁     r = 0 '   n δ n − r ⊗ n r .

∎

5. Conclusions

It can be concluded that the permanent of a triangular matrix and a diagonal strictly double ℝ - 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.

 Data availability statement

No data are associated with this article.

 Acknowledgements

The author would like to thank Universitas Sebelas Maret and the students who are members of the algebra research team for their joint discussions.

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