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*Corresponding Author: Hanifa Zekraoui, Email: hanifazekraoui@yahoo.fr
RESEARCH ARTICLE
Available Online at
Asian Journal of Mathematical Sciences 2017; 1(1):1-6
www.ajms.in
Generalized inverses of matrices in Max-Plus-Algebra
1
Hanifa Zekraoui*, 2
Cenap Ozel , 3
Beyhan Yeni
1
*Department of Mathematics, Oum-El-Bouaghi University, 04000, Algeria.
2
Department of Mathematics, Dokuz Eylul University, Izmir, Turkey.
Department of Mathematics, king Abd Al Aziz University, KSA
3
Department of Mathematics, Dokuz Eylul University, Izmir, Turkey.
Received on: 31/01/2017, Revised on: 15/02/2017, Accepted on: 20/02/2017
ABSTRACT
The aim of this paper is to study the generalized inverses of a matrix in Max-plus Algebra and examine
the analogy with some remarkable properties of them in classical Matrix Theory (in Linear Algebra).
Because of the complexity of computation on matrices with high sizes, we will deal with the 2 by 2 and 3
by 3 matrices only and using them in solving the matrix equations.
Key words: Max-Plus-Algebra, matrix operation, matrix equation, discrepancy matrix, generalized
inverse, Shur complement, MSC [2010]: 53C25, 83C05, 57N16, 15A09, 15A21.
INTRODUCTION
In Linear Algebra, we have the matrix equation Ax = y has a solution if and only if y ∈ R(A). Therefore, it
will be in the form x = A−
y, where A−
is a generalized inverse of A. There is a form for the general solution
( by replacing the chosen A−
by the general form which gives all g-inverses and so all solutions ), see 1
.
Many properties of generalized inverses of matrices took place in 2,3
. A Max-Plus Algebra is a semi-ring
over the union of real numbers and ε = −∞, equipped with maximum and addition as the two binary
operations. The zero and the unit of this semi-ring is ε and 0 respectively. The sum and the product of two
matrices A and B in Max-Plus Algebra are exactly defined as the sum and the product of them in Linear
Algebra, by replacing addition and multiplication operations by max and addition operations denoted by
A ⊕B and A ⊗
B. The zero matrix O and the identity I are respectively, the matrix with−∞entries and the
diagonal matrix with zero entries on the diagonal and−∞otherwise. As the role of generalized inverses in
Linear Algebra and optimization, the Max-Plus Algebra uses the operation of taking a maximum, thus
making it an ideal candidate for mathematically describing problems in operations research. Most often,
problems in Operations Research have been solved by the development of algorithmic procedures that
lead to optimal solutions. Solving the matrix equation Ax = y in Max PlusAlgebra (when there is a
solution) by using the concept of generalized inverses of matrices will be an original research. The aim of
this paper is to study the generalized inverses of a matrix in Max-plus-Algebra and examine the analogy
with some remarkable properties of them in classical Matrix Theory (in Linear Algebra). Because of the
complexity of computation on matrices with high sizes, we will deal with the 2 by 2 and 3 by 3 matrices
only. The paper is divided in three parts; the first one is destined for the 2 by 2 matrices while the second
one treats the 3 by 3 matrices. The last part checks the existence of the Moore Penrose inverse. If the
Moore Penrose inverse exists, then it is unique. This result can be found in any literature, for instance,
one can see [1], or prove it directly by the use of the four equations in the definition of the Moore
Penrose.
In Linear Algebra, a square matrix A is invertible if and only if there exists a matrix B satisfying AB = BA
= I or equivalently to detA 6= 0, otherwise, there exists what is called a generalized inverse of a matrix A,
it is a matrix X satisfying AXA = A. it is often called a {1}- inverse. A {2}- inverse of A is a matrix X
Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra
2
© 2017, AJMS. All Rights Reserved.
satisfying XAX = X. If the matrix X is both {1}- inverse and {2}- inverse, then it called a {1,2}- inverse
and often denoted by A(1,2)
. Using the analogy of this definition we define the generalized inverses of a
matrix A in Max-Plus Algebra with respect to the notations above, i.e. A ⊗
X ⊗
A = A and X ⊗
A ⊗
X = X.
The following example shows that an invertible matrix in classical matrix theory doesn’t necessarily
invertible in Max-Plus Algebra.
Example 1 Let . If there exist such that
AB = I, then we get the folowing system
max(a,c −1) =0 and max(a + 2,c) = − ∞
max(b,d −1) =−∞ and max(b + 2,d) = 0
The first equation gives
(a = 0 and c ≤ 1 or c = 1 and a ≤ 0) and (a = c = −∞)
which is impossible to hold. Then A−1
doesn’t exist in Max-Plus Algebra, how-ever in
classical matrix theory.
Example 2 Now, we try to see if a g-inverse of A in the previous example
exists. Let , such that A ⊗
B ⊗
A = A.
Then, we get the following system :
max(max(a,c −1),max(b,d −1) + 2) = 0 (1)
max(max(a + 2,c),max(b + 2,d) + 2) = 2
max(max(a,c −1) −1,max (b,d −1)) = −1
max(max(a + 2,c) −1,max(b + 2,d)) = 0,
which is equivalent to the following system:
max(a,c −1) ≤ 0 , max(b,d −1) ≤ −2,
max(a + 2,c) ≤ 2, max(b + 2,d) ≤ 0,
max(a,c −1) ≤ 0, max(b,d −1) ≤ −1,
max(a + 2,c) ≤ 1, max(b + 2,d) ≤ 0.
From the first equation in 1, we get
a ≤ 0, c ≤ 1, b ≤−2, d ≤−1
We remark that if we take, equality for a and c,then the third inequality in (1) can’t hold because we get
max(a + 2,c) = 2 6= 1.So the solutions are-
a <0, c <1, b ≤−2, d ≤−1,
which means that there are infinitely many g-inverses of . How-ever this matrix has only
one inverse equals to A−1
in classical matrix theory.
Part I
The generalized inverses of 2 by 2 matrix
2A {1}-inverse of a square matrix
Let and , such that A ⊗
X ⊗
A = A. Then,
we get the following system
max(max(a + x1,b + x3) + a,max(a + x2,b + x4) + c) = a (2)
max(max(a + x1,b + x3) + b,max(a + x2,b + x4) + d) = b
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Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra
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max(max(c + x1,d + x3) + a,max(c + x2,d + x4) + c) = c
max(max(c + x1,d + x3) + b,max(c + x2,d + x4) + d) = d
Equations in system 2 respectively yield
x1 ≤ −a, x2 ≤−c, x3 ≤−b, x4 ≤ a −b −c x1 ≤ −a, x2 ≤ b −d −a, x3 ≤−b, x4 ≤−d x1 ≤ −a, x2 ≤−c, x3 ≤ c −a −d, x4
≤−d x1 ≤ d −b −c, x2 ≤−c, x3 ≤−b, x4 ≤−d
which give
x1≤min(-a,d –b –c), x2 ≤ min(−c,b −d −a), x3 ≤ min(−b,c −a −d), x4 ≤ min(−d,a −b −c) (3)
Thus, we can confirm the following proposition:
Proposition 3 Every 2 by 2 matrix has infinitely many {1}-inverses in MaxPlus Algebra.
3 A {2}-inverse of a square matrix
Let and , such that X ⊗
A ⊗
X = X. Then,
We get the following system
max(max(x1 + a,x2 + c) + x1,max(x1 + b,x2 + d) + x3) = x1
max(max(x1 + a,x2 + c) + x2,max(x1 + b,x2 + d) + x4) =x2
max(max(x3 + a,x4 + c) + x1,max(x3 + b,x4 + d) + x3) =x3
max(max(x3 + a,x4 + c) + x2,max(x3 + b,x4 + d) + x4) =x4
Equations in system 4 respectively yield
x1 ≤ −a, x2 ≤−c, x3 ≤−b x1 ≤ −a, x2 ≤−c, x4 ≤−d, x4 ≤ x2 −x1 −b x1 ≤ −a, x4 ≤ x3 −x1 −c, x3 ≤−b, x4 ≤−d x2 ≤ −c,
x3 ≤−b, x2 + x3 + a ≤ x4 ≤−d
which give
x1 ≤−a, x2 ≤−c, x3 ≤−b, x2+x3+a ≤ x4 ≤ min(−d,x2 −x1 −b,x3 −x1 −c). (5)
Thus, we have the following proposition:
Proposition 4 Every 2 by 2 matrix has infinitely many {2}-inverses in MaxPlus Algebra.
If we examine ralations in (3) and (5) , we remark that, which make a {1}inverse not to be a {2}-inverse
is its last entry. Hence, we have the following proposition:
Proposition 5 Let and . Then X is a {1,2}-inverse of A if and only if
x1 ≤−a, x2 ≤−c, x3 ≤−b and x2+x3+a ≤ x4 ≤ min(−d,a −b −c,x2 −x1 −b,x3 −x1 −c). (6)
If we examine the previous situation and the result in 6 , we find an analogy with the results in classical
matrix theory:
If we take a block matrix M= , such that A invertible, then a {1}-inverse of ,
the such generalized inverse to be a {2}-inverse, is that the last block Z ( entry for the case 2 by 2 matrix
has to satisfy
Z = Y AX.
Now, comparing this result with x4 in 6, we find the analogy:
AJMS,
Jan-Feb,
2017,
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Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra
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© 2017, AJMS. All Rights Reserved.
x3 + a + x2 = x2 + x3 + a ≤ x4.
The last minimization of x4 in 6 has exactly the analogy with Shur complement
in classical matrix theory: If , such that A is invertible of order r = rank (M), then M can be
represented in the form:
(i.e. D = CA−1
B), then a {1,2}-inverse of M is
such that represent a {1,2}-inverse of CA−1
B. In the case of 2 by 2 matrix, A, B and C are reals,
then which exactly analogues to
−b+a−c. If A −1
does not exist in
max-Plus-Algebra, then we replace it by a {1}-inverse of A as this one always exists.
Part II
The generalized inverse of a 3 by 3 matrix
Let and , such that A⊗
X⊗
A = A. Then, the first column of
this product is
By the same way, we obtain the remaining columns, which give the explicit form; for all k and l in the
index set {1,2,3}
(7)
From 7, for all k and l in the index set {1,2,3}, we get the solutions xji in the form xji ≤ min(akl −ail −akj)
for all j, i = 1,2,3
Then, we have the following proposition
Proposition 6 Every 3 by 3 matrix has infinitely many {1}-inverses in MaxPlus Algebra.
Part III
The Moore Penrose invrse
We recall that the moore Penrose inverse of a matrix A is the matrix denoted
A+
satisfying the four equations A ⊗
A+
⊗
A = A, A+
⊗
A ⊗
A+
= A+
, (A ⊗
A+
)∗
= A ⊗
A+
and (A+
⊗
A)∗
= A+
⊗
A.
The existence of the Moore penrose inverse in Linear Algebra depends of the vector space we work in, If
the vector space is over field of characteristic zero such R or C, with an inner product, then the Moore
Penrose inverse always exists. If the vector space is over a finite field, or without an inner product such
Minkovski space, then there are conditions of existence. For this subject, one can refer to [5], [6], [7].
For the reason of calculation, in this paper, we will look for the Moore Penrose inverse of 2 by 2 matrix,
just to have an idea.
AJMS,
Jan-Feb,
2017,
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Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra
5
© 2017, AJMS. All Rights Reserved.
Let us use results in 6 of Proposition 5 and we examine the third and the fourth equations to extract our
conditions.
Let , such that (A ⊗
A+
)∗
= A ⊗
A+
, then we get max(x1 + c,x3 + d) = max(x2 + a,x4 + b) ,
which gives
x1 −x2 = a −c or, x1 −x4 = b −c or, x2 −x3 = d −a or, x3 −x4 = b –d (8)
Let in addition (A+
⊗
A)∗
= A+
⊗
A, then we get max(x1 + b,x2 + d) = max(x3 + a,x4 + c) ,
which gives
x1 −x3 = a −b or, x1 −x4 = c −b or, x2 −x3 = a −d or, x2 −x4 = c −d (9)
The existence of the Moore penrose inverse of A depends on the existence of the solutions of the system
of inqualities in 6, 8 and 9. Hence, by 6 and from 8, suppose that x1 −x2 = a −c, then we have
x1 ≤ −a −c ≥ x2 = x1 −a + c ≥ 2x1 +c
which gives x1 ≤ min(−a,−c).
By 6 and from 9, suppose that x1 −x3 = a −b, then
x1≤ −a
−b≥ x3 = x1 −a + b ≥ 2x1 + b
which yields
x1 ≤ min(−a,−b)
Hence,
x1 ≤ min(−a,−b,−c).
By 6 and the previous assumptions, we get
2x1 −a + b + c ≤ x4 ≤ min(−d,a −b −c,c −a −b,b −c −a).
Proposition 7 Let and , such that
x1 ≤ min(−a,−b,−c),
x2 = x1 −a + c ≤−c,
x3 = x1 −a + b ≤−b,
2x1 −a + b + c ≤ x4 ≤ min(−d,a −b −c,c −a −b,b −c −a),
then X = A+
.
REFERENCES
1. H. Zekraoui, Propri´et´es Alg´ebriques des G k inverses des Matrices, Th`ese de Doctorat en
Sciences, universit´e Batna, 2011.
2. H. Zekraoui and S. Guedjiba, On Algebraic Properties of Generalized Inverses of Matrices,
International Journal of Algebra, Vol. 2, 2008, no. 13 , 633 – 643
3. H. Zekraoui and S.Guedjiba, Semigroup of generalized inverses of matrices, Applied Sciences,
Vol.12, 2010, pp. 146-152.
4. Samuel Asante Gyamerah, Peter Kwaku Boateng and Prince Harvim, Maxplus Algebra and
Application to Matrix Operations, British Journal of Mathematics & Computer Science, 12(3),
(2015), 1-14.
5. H. Zekraoui, Z. Al-Zhour and C. Ozel, Some New Algebraic and Topological Properties of the
Minkowski Inverse in the Minkowski Space, Hindawi Publishing Corporation, The Scientific
World Journal, Article ID 765732 , (2013), 6 pages.
6. H. Zekraoui, C. Ozel, Generalized inverses of matrices and Least Squares Solutions in linear
codes, The conference CITCEP (2015)
7. D. Fulton ,Generalized inverse of matrices over finite field, Discretemathematics, 21, ( 1978), 23-
29.
AJMS,
Jan-Feb,
2017,
Vol.
1,
Issue
1
Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra
6
© 2017, AJMS. All Rights Reserved.
8. H. Zekraoui, Propri´eties algebriques des Gk
−inverses des matrices, th`ese de Doctorat,
Universit´e de Batna, (2011).
AJMS,
Jan-Feb,
2017,
Vol.
1,
Issue
1

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1 Hanifa Zekraoui.pdf

  • 1. *Corresponding Author: Hanifa Zekraoui, Email: hanifazekraoui@yahoo.fr RESEARCH ARTICLE Available Online at Asian Journal of Mathematical Sciences 2017; 1(1):1-6 www.ajms.in Generalized inverses of matrices in Max-Plus-Algebra 1 Hanifa Zekraoui*, 2 Cenap Ozel , 3 Beyhan Yeni 1 *Department of Mathematics, Oum-El-Bouaghi University, 04000, Algeria. 2 Department of Mathematics, Dokuz Eylul University, Izmir, Turkey. Department of Mathematics, king Abd Al Aziz University, KSA 3 Department of Mathematics, Dokuz Eylul University, Izmir, Turkey. Received on: 31/01/2017, Revised on: 15/02/2017, Accepted on: 20/02/2017 ABSTRACT The aim of this paper is to study the generalized inverses of a matrix in Max-plus Algebra and examine the analogy with some remarkable properties of them in classical Matrix Theory (in Linear Algebra). Because of the complexity of computation on matrices with high sizes, we will deal with the 2 by 2 and 3 by 3 matrices only and using them in solving the matrix equations. Key words: Max-Plus-Algebra, matrix operation, matrix equation, discrepancy matrix, generalized inverse, Shur complement, MSC [2010]: 53C25, 83C05, 57N16, 15A09, 15A21. INTRODUCTION In Linear Algebra, we have the matrix equation Ax = y has a solution if and only if y ∈ R(A). Therefore, it will be in the form x = A− y, where A− is a generalized inverse of A. There is a form for the general solution ( by replacing the chosen A− by the general form which gives all g-inverses and so all solutions ), see 1 . Many properties of generalized inverses of matrices took place in 2,3 . A Max-Plus Algebra is a semi-ring over the union of real numbers and ε = −∞, equipped with maximum and addition as the two binary operations. The zero and the unit of this semi-ring is ε and 0 respectively. The sum and the product of two matrices A and B in Max-Plus Algebra are exactly defined as the sum and the product of them in Linear Algebra, by replacing addition and multiplication operations by max and addition operations denoted by A ⊕B and A ⊗ B. The zero matrix O and the identity I are respectively, the matrix with−∞entries and the diagonal matrix with zero entries on the diagonal and−∞otherwise. As the role of generalized inverses in Linear Algebra and optimization, the Max-Plus Algebra uses the operation of taking a maximum, thus making it an ideal candidate for mathematically describing problems in operations research. Most often, problems in Operations Research have been solved by the development of algorithmic procedures that lead to optimal solutions. Solving the matrix equation Ax = y in Max PlusAlgebra (when there is a solution) by using the concept of generalized inverses of matrices will be an original research. The aim of this paper is to study the generalized inverses of a matrix in Max-plus-Algebra and examine the analogy with some remarkable properties of them in classical Matrix Theory (in Linear Algebra). Because of the complexity of computation on matrices with high sizes, we will deal with the 2 by 2 and 3 by 3 matrices only. The paper is divided in three parts; the first one is destined for the 2 by 2 matrices while the second one treats the 3 by 3 matrices. The last part checks the existence of the Moore Penrose inverse. If the Moore Penrose inverse exists, then it is unique. This result can be found in any literature, for instance, one can see [1], or prove it directly by the use of the four equations in the definition of the Moore Penrose. In Linear Algebra, a square matrix A is invertible if and only if there exists a matrix B satisfying AB = BA = I or equivalently to detA 6= 0, otherwise, there exists what is called a generalized inverse of a matrix A, it is a matrix X satisfying AXA = A. it is often called a {1}- inverse. A {2}- inverse of A is a matrix X
  • 2. Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra 2 © 2017, AJMS. All Rights Reserved. satisfying XAX = X. If the matrix X is both {1}- inverse and {2}- inverse, then it called a {1,2}- inverse and often denoted by A(1,2) . Using the analogy of this definition we define the generalized inverses of a matrix A in Max-Plus Algebra with respect to the notations above, i.e. A ⊗ X ⊗ A = A and X ⊗ A ⊗ X = X. The following example shows that an invertible matrix in classical matrix theory doesn’t necessarily invertible in Max-Plus Algebra. Example 1 Let . If there exist such that AB = I, then we get the folowing system max(a,c −1) =0 and max(a + 2,c) = − ∞ max(b,d −1) =−∞ and max(b + 2,d) = 0 The first equation gives (a = 0 and c ≤ 1 or c = 1 and a ≤ 0) and (a = c = −∞) which is impossible to hold. Then A−1 doesn’t exist in Max-Plus Algebra, how-ever in classical matrix theory. Example 2 Now, we try to see if a g-inverse of A in the previous example exists. Let , such that A ⊗ B ⊗ A = A. Then, we get the following system : max(max(a,c −1),max(b,d −1) + 2) = 0 (1) max(max(a + 2,c),max(b + 2,d) + 2) = 2 max(max(a,c −1) −1,max (b,d −1)) = −1 max(max(a + 2,c) −1,max(b + 2,d)) = 0, which is equivalent to the following system: max(a,c −1) ≤ 0 , max(b,d −1) ≤ −2, max(a + 2,c) ≤ 2, max(b + 2,d) ≤ 0, max(a,c −1) ≤ 0, max(b,d −1) ≤ −1, max(a + 2,c) ≤ 1, max(b + 2,d) ≤ 0. From the first equation in 1, we get a ≤ 0, c ≤ 1, b ≤−2, d ≤−1 We remark that if we take, equality for a and c,then the third inequality in (1) can’t hold because we get max(a + 2,c) = 2 6= 1.So the solutions are- a <0, c <1, b ≤−2, d ≤−1, which means that there are infinitely many g-inverses of . How-ever this matrix has only one inverse equals to A−1 in classical matrix theory. Part I The generalized inverses of 2 by 2 matrix 2A {1}-inverse of a square matrix Let and , such that A ⊗ X ⊗ A = A. Then, we get the following system max(max(a + x1,b + x3) + a,max(a + x2,b + x4) + c) = a (2) max(max(a + x1,b + x3) + b,max(a + x2,b + x4) + d) = b AJMS, Jan-Feb, 2017, Vol. 1, Issue 1
  • 3. Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra 3 © 2017, AJMS. All Rights Reserved. max(max(c + x1,d + x3) + a,max(c + x2,d + x4) + c) = c max(max(c + x1,d + x3) + b,max(c + x2,d + x4) + d) = d Equations in system 2 respectively yield x1 ≤ −a, x2 ≤−c, x3 ≤−b, x4 ≤ a −b −c x1 ≤ −a, x2 ≤ b −d −a, x3 ≤−b, x4 ≤−d x1 ≤ −a, x2 ≤−c, x3 ≤ c −a −d, x4 ≤−d x1 ≤ d −b −c, x2 ≤−c, x3 ≤−b, x4 ≤−d which give x1≤min(-a,d –b –c), x2 ≤ min(−c,b −d −a), x3 ≤ min(−b,c −a −d), x4 ≤ min(−d,a −b −c) (3) Thus, we can confirm the following proposition: Proposition 3 Every 2 by 2 matrix has infinitely many {1}-inverses in MaxPlus Algebra. 3 A {2}-inverse of a square matrix Let and , such that X ⊗ A ⊗ X = X. Then, We get the following system max(max(x1 + a,x2 + c) + x1,max(x1 + b,x2 + d) + x3) = x1 max(max(x1 + a,x2 + c) + x2,max(x1 + b,x2 + d) + x4) =x2 max(max(x3 + a,x4 + c) + x1,max(x3 + b,x4 + d) + x3) =x3 max(max(x3 + a,x4 + c) + x2,max(x3 + b,x4 + d) + x4) =x4 Equations in system 4 respectively yield x1 ≤ −a, x2 ≤−c, x3 ≤−b x1 ≤ −a, x2 ≤−c, x4 ≤−d, x4 ≤ x2 −x1 −b x1 ≤ −a, x4 ≤ x3 −x1 −c, x3 ≤−b, x4 ≤−d x2 ≤ −c, x3 ≤−b, x2 + x3 + a ≤ x4 ≤−d which give x1 ≤−a, x2 ≤−c, x3 ≤−b, x2+x3+a ≤ x4 ≤ min(−d,x2 −x1 −b,x3 −x1 −c). (5) Thus, we have the following proposition: Proposition 4 Every 2 by 2 matrix has infinitely many {2}-inverses in MaxPlus Algebra. If we examine ralations in (3) and (5) , we remark that, which make a {1}inverse not to be a {2}-inverse is its last entry. Hence, we have the following proposition: Proposition 5 Let and . Then X is a {1,2}-inverse of A if and only if x1 ≤−a, x2 ≤−c, x3 ≤−b and x2+x3+a ≤ x4 ≤ min(−d,a −b −c,x2 −x1 −b,x3 −x1 −c). (6) If we examine the previous situation and the result in 6 , we find an analogy with the results in classical matrix theory: If we take a block matrix M= , such that A invertible, then a {1}-inverse of , the such generalized inverse to be a {2}-inverse, is that the last block Z ( entry for the case 2 by 2 matrix has to satisfy Z = Y AX. Now, comparing this result with x4 in 6, we find the analogy: AJMS, Jan-Feb, 2017, Vol. 1, Issue 1
  • 4. Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra 4 © 2017, AJMS. All Rights Reserved. x3 + a + x2 = x2 + x3 + a ≤ x4. The last minimization of x4 in 6 has exactly the analogy with Shur complement in classical matrix theory: If , such that A is invertible of order r = rank (M), then M can be represented in the form: (i.e. D = CA−1 B), then a {1,2}-inverse of M is such that represent a {1,2}-inverse of CA−1 B. In the case of 2 by 2 matrix, A, B and C are reals, then which exactly analogues to −b+a−c. If A −1 does not exist in max-Plus-Algebra, then we replace it by a {1}-inverse of A as this one always exists. Part II The generalized inverse of a 3 by 3 matrix Let and , such that A⊗ X⊗ A = A. Then, the first column of this product is By the same way, we obtain the remaining columns, which give the explicit form; for all k and l in the index set {1,2,3} (7) From 7, for all k and l in the index set {1,2,3}, we get the solutions xji in the form xji ≤ min(akl −ail −akj) for all j, i = 1,2,3 Then, we have the following proposition Proposition 6 Every 3 by 3 matrix has infinitely many {1}-inverses in MaxPlus Algebra. Part III The Moore Penrose invrse We recall that the moore Penrose inverse of a matrix A is the matrix denoted A+ satisfying the four equations A ⊗ A+ ⊗ A = A, A+ ⊗ A ⊗ A+ = A+ , (A ⊗ A+ )∗ = A ⊗ A+ and (A+ ⊗ A)∗ = A+ ⊗ A. The existence of the Moore penrose inverse in Linear Algebra depends of the vector space we work in, If the vector space is over field of characteristic zero such R or C, with an inner product, then the Moore Penrose inverse always exists. If the vector space is over a finite field, or without an inner product such Minkovski space, then there are conditions of existence. For this subject, one can refer to [5], [6], [7]. For the reason of calculation, in this paper, we will look for the Moore Penrose inverse of 2 by 2 matrix, just to have an idea. AJMS, Jan-Feb, 2017, Vol. 1, Issue 1
  • 5. Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra 5 © 2017, AJMS. All Rights Reserved. Let us use results in 6 of Proposition 5 and we examine the third and the fourth equations to extract our conditions. Let , such that (A ⊗ A+ )∗ = A ⊗ A+ , then we get max(x1 + c,x3 + d) = max(x2 + a,x4 + b) , which gives x1 −x2 = a −c or, x1 −x4 = b −c or, x2 −x3 = d −a or, x3 −x4 = b –d (8) Let in addition (A+ ⊗ A)∗ = A+ ⊗ A, then we get max(x1 + b,x2 + d) = max(x3 + a,x4 + c) , which gives x1 −x3 = a −b or, x1 −x4 = c −b or, x2 −x3 = a −d or, x2 −x4 = c −d (9) The existence of the Moore penrose inverse of A depends on the existence of the solutions of the system of inqualities in 6, 8 and 9. Hence, by 6 and from 8, suppose that x1 −x2 = a −c, then we have x1 ≤ −a −c ≥ x2 = x1 −a + c ≥ 2x1 +c which gives x1 ≤ min(−a,−c). By 6 and from 9, suppose that x1 −x3 = a −b, then x1≤ −a −b≥ x3 = x1 −a + b ≥ 2x1 + b which yields x1 ≤ min(−a,−b) Hence, x1 ≤ min(−a,−b,−c). By 6 and the previous assumptions, we get 2x1 −a + b + c ≤ x4 ≤ min(−d,a −b −c,c −a −b,b −c −a). Proposition 7 Let and , such that x1 ≤ min(−a,−b,−c), x2 = x1 −a + c ≤−c, x3 = x1 −a + b ≤−b, 2x1 −a + b + c ≤ x4 ≤ min(−d,a −b −c,c −a −b,b −c −a), then X = A+ . REFERENCES 1. H. Zekraoui, Propri´et´es Alg´ebriques des G k inverses des Matrices, Th`ese de Doctorat en Sciences, universit´e Batna, 2011. 2. H. Zekraoui and S. Guedjiba, On Algebraic Properties of Generalized Inverses of Matrices, International Journal of Algebra, Vol. 2, 2008, no. 13 , 633 – 643 3. H. Zekraoui and S.Guedjiba, Semigroup of generalized inverses of matrices, Applied Sciences, Vol.12, 2010, pp. 146-152. 4. Samuel Asante Gyamerah, Peter Kwaku Boateng and Prince Harvim, Maxplus Algebra and Application to Matrix Operations, British Journal of Mathematics & Computer Science, 12(3), (2015), 1-14. 5. H. Zekraoui, Z. Al-Zhour and C. Ozel, Some New Algebraic and Topological Properties of the Minkowski Inverse in the Minkowski Space, Hindawi Publishing Corporation, The Scientific World Journal, Article ID 765732 , (2013), 6 pages. 6. H. Zekraoui, C. Ozel, Generalized inverses of matrices and Least Squares Solutions in linear codes, The conference CITCEP (2015) 7. D. Fulton ,Generalized inverse of matrices over finite field, Discretemathematics, 21, ( 1978), 23- 29. AJMS, Jan-Feb, 2017, Vol. 1, Issue 1
  • 6. Hanifa Zekraoui et al./ Generalized inverses of matrices in Max-Plus-Algebra 6 © 2017, AJMS. All Rights Reserved. 8. H. Zekraoui, Propri´eties algebriques des Gk −inverses des matrices, th`ese de Doctorat, Universit´e de Batna, (2011). AJMS, Jan-Feb, 2017, Vol. 1, Issue 1