Generalized inverses of matrices in Max-Plus-Algebra Hanifa Zekraoui Departement of Mathematics, Oum-El-Bouaghi University, 04000, Algeria.
[email protected] Cenap Ozel Departement of Mathematics, Dokuz Eylul University, Izmir, Turkey. Departement of Mathematics, king Saud University, KSA
[email protected] October 13, 2016 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.
1
Introduction
In Linear Algebra, we have the matrix equation Ax = y has a solution if and only if y 2 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 " = 1, equipped with maximum and addition as the two binary operations. The zero and the unit of this semi-ring is " and 0 respectively. The sum and the product of two matrices A and B in Max-Plus Algebra are exactly de…ned 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
1
respectively, the matrix with 1 entries and the diagonal matrix with zero entries on the diagonal and 1 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 …rst one is destined for the 2 by 2 matrices while the second one treats the 3 by 3 matrices. The last part focused on solving the matrix equation Ax = y by using the form x = A y and proving that the condition of existence of the solution cited above is analogous to the condition on the discrepancy matrix or the reduced discrepancy matrix described in [4]. 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 det A 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 f1g- inverse. A f2g- inverse of A is a matrix X satisfying XAX = X. If the matrix X is both f1g- inverse and f2g- inverse, then it called a f1; 2g- inverse and often denoted by A(1;2) . Using the analogy of this de…nition we de…ne 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. 0 1 . If there exist B = 2 0 AB = I, then we get the folowing system
Example 1 Let A =
max (a; c max (b; d
a b c d
such that
1) = 0 and max (a + 2; c) = 1 1) = 1 and max (b + 2; d) = 0
The …rst equation gives (a = 0 and c
1 or c = 1 and a
0) and (a = c =
1)
which is impossible to hold. Then A 1 doesn’t exist in Max-Plus Algebra, how1 0 2 in classical matrix theory. ever A 1 = 1 0 Example 2 Now, we try to see if a g-inverse of A in the previous example
2
exists. Let B =
a b c d
, such that A
B
A = A.
max (max (a; c 1) ; max (b; d 1) + 2) max (max (a; c 1) max (max (a + 2; c) ; max (b + 2; d) + 2) max (max (a + 2; c)
1; max (b; d 1)) 1; max (b + 2; d))
Then, we get the following system : max (max (a; c 1) ; max (b; d 1) + 2) = 0 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) max (a + 2; c) max (a; c 1) max (a + 2; c)
0 , max (b; d 1) 2, 2, max (b + 2; d) 0 0, max (b; d 1) 1, 1, max (b + 2; d) 0.
(1)
From the …rst 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, 0 1 . How2 0 in classical matrix theory.
which means that there are in…nitely many g-inverses of A = ever this matrix has only one inverse equals to A
1
Part I
The generalized inverses of 2 by 2 matrix 2
A f1g-inverse of a square matrix
a b and X = c d we get the following system
Let A =
x1 x3
x2 x4
3
, such that A
X
A = A. Then,
=
0 2
1 0
max (max (a + x1 ; b + x3 ) + a; max (a + x2 ; b + x4 ) + c) = a max (max (a + x1 ; b + x3 ) + b; max (a + x2 ; b + x4 ) + d) = b 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
(2)
Equations in system 2 respectively yield x1 x1 x1 x1
a, x2 c, x3 b, x4 a b a, x2 b d a, x3 b, x4 a, x2 c, x3 c a d, x4 d b c, x2 c, x3 b, x4
c d d d
which give x1 x3
min ( a; d min ( b; c
b a
c) , x2 d) , x4
min ( c; b min ( d; a
d b
a) , c)
(3)
Thus, we can con…rm the following proposition: Proposition 3 Every 2 by 2 matrix has in…nitely many f1g-inverses in MaxPlus Algebra.
3
A f2g-inverse of a square matrix
a b and X = c d we get the following system
Let A =
x1 x3
x2 x4
, such that X
A
X = X. Then,
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
(4)
Equations in system 4 respectively yield x1 x1 x1 x2
a, x2 a, x2 a, x4 c, x3
c, x3 b c, x4 d, x4 x3 x1 c, x3 b, x2 + x3 + a
x2 x1 b, x4 x4 d
b d
which give x1
a, x2
c, x3
b, x2 +x3 +a
x4
Thus, we have the following proposition: 4
min ( d; x2
x1
b; x3
x1
c) . (5)
Proposition 4 Every 2 by 2 matrix has in…nitely many f2g-inverses in MaxPlus Algebra. If we examine ralations in 3 and 5 , we remark that, which make a f1ginverse not to be a f2g-inverse is its last entry. Hence, we have the following proposition: a b and X = c d Then X is a f1; 2g-inverse of A if and only if
Proposition 5 Let A =
x1
a, x2
c, x3
b and x2 +x3 +a
x1 x3
x4
x2 x4
.
min ( d; a
b
c; x2
x1 b; x3 (6)
If we examine the previous situation and the result in 6 , we …nd an analogy with the results in classical matrix theory: If we take a block matrix M = A 0 A 1 X , such that A invertible, then a f1g-inverse of M is , 0 0 Y Z the such generalized inverse to be a f2g-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 …nd the analogy: x3 + a + x2 = x2 + x3 + a
x4 .
The last minimization of x4 in 6 has exactly the analogy with Shur complement A B in classical matrix theory: If M = , such that A is invertible of order C D r = rank (M ), then M can be represented in the form: M=
A C
(i.e. D = CA
B D 1
Ir CA
1
0 I
A 0 0 CA 1 B
Ir 0
A
1
B
I
B), then a f1; 2g-inverse of M is A 1B I
Ir 0 such that CA
=
1
B
A 1 0
0 CA
1
Ir CA
B
represent a f1; 2g-inverse of CA
1
1
0 I
B. In the case of 2 by 1
2 matrix, A, B and C are reals, then CA 1 B = CA 1 B = AB 1 C 1 which exactly analogs to a b c. If A 1 does not exist in max-Plus-Algebra, then we replace it by a f1g-inverse of A as this one always exists.
5
x1
c) .
Part II
The generalized inverse of a 3 by 3 matrix 0
1 0 a11 a12 a13 x11 Let A = @ a21 a22 a23 A and X = @ x21 a31 a32 a33 x31 A = A. Then, the …rst column of this product
x12 x22 x32 is
1 x13 x23 A, such that A X x33
3
3
3
j=1
j=1
j=1
max a11 + max (a1j + xj1 ) ; a21 + max (a1j + xj2 ) ; a31 + max (a1j + xj3 ) 3
3
3
j=1
j=1
j=1
max a11 + max (a2j + xj1 ) ; a21 + max (a2j + xj2 ) ; a31 + max (a2j + xj3 ) 3
3
3
j=1
j=1
j=1
max a11 + max (a3j + xj1 ) ; a21 + max (a3j + xj2 ) ; a31 + max (a3j + xj3 )
3
3
i=1
j=1
3
3
i=1
j=1
3
3
i=1
j=1
= max ai1 + max (a3j + xji )
(7)
From 7, for all k and l in the index set f1; 2; 3g, 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 in…nitely many f1g-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 …eld of characteristic zero such R or C, with an inner product, then the Moore Penrose inverse always exists. If the vector space is
6
3
j=1
= max ai1 + max (a2j + xji )
By the same way, we obtain the remaining columns, which give the explicit form; for all k and l in the index set f1; 2; 3g akl = max ail + max (akj + xji )
3
i=1
= max ai1 + max (a1j + xji )
over a …nite …eld, 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. Let us use results in 6 of Proposition 5 and we examine the third and the fourth equations to extract our conditions. x1 x2 Let A+ = , such that (A A+ ) = A A+ , then we get x3 x4 max (x1 + c; x3 + d) = max (x2 + a; x4 + b) , which gives x1
x2 = a
c or, x1
Let in addition (A+
x4 = b
c or, x2
A) = A+
x3 = d
a or, x3
x4 = b
d (8)
x4 = c
d (9)
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
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 c
a x2 = x1
a+c
2x1 + c
which gives x1
min ( a; c) .
By 6 and from 9, suppose that x1 x1 b
x3 = a
a x3 = x1
b, then
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
7
b
c; c
a
b; b
c
a) .
Proposition 7 Let A =
2x1
a b c d
and X =
x1 min ( a; b; c) , x2 = x1 a + c c, x3 = x1 a + b b, a+b+c x4 min ( d; a b
x1 x3
c; c
x2 x4
, such that
a
b; b
c
a) ,
then X = A+ . The proof of uniqueness of the Moore Penrose inverse in Linear Algebra depends only on the four equations and does not include any of the properties of the vector space, one can refer to [8]. In Max-Plus-Algebra, from the previous results , it seems we can have more than one, how to solve this problem? is an open question for the readers that we wish to suggest us some ideas.
Part IV
Solving the matrix equation Ax = y by generalized inverses in Max-Plus-Algebra 4
future research
Extending notions of types of generalized inverses like Moore-Penrose, Least squares, Minimum Norm, group inverse, projectors, ...etc and their properties in Linear Algebra to Max-Plus-Algebra.
References [1] H. Zekraoui, Propriétés Algébriques des G_k inverses des Matrices, Thèse de Doctorat en Sciences, université 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. 8
[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 Scienti…c 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 …nite …eld, Discretemathematics, 21, ( 1978), 23-29. [8] H. Zekraoui, Propriéties algebriques des Gk inverses des matrices, thèse de Doctorat, Université de Batna, (2011).
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