The rank of a fuzzy matrix and its evaluation

Fuzzy Sets and Systems 38 (1990) 355-364 North-Holland

355

THE R A N K OF A F U Z Z Y M A T R I X A N D ITS EVALUATION S. KAGUEI and A. OHSATO* Division of Electrical and Computer Engineering, Faculty of Engineering, Yokohama National University, Yokohama 240, Japan Received January 1989 Revised April 1989

Abstract: A new type of matrix rank, which is called margin rank in this article, is introduced to a fuzzy matrix defined to be rectangular array of fuzzy numbers. The new rank is, in general, a real number and consistent with the conventional integer-valued rank, defined for the crisp matrix. The margin rank indicates the margin of retaining the rank of the mean matrix, which enables us to represent the grade of some characteristics described by the ordinary rank of a matrix. In this article the definition of the new rank and a procedure for its evaluation are shown with several examples.

Keywords: Algebra; fuzzy matrix; rank of matrix.

1. Introduction The rank of a matrix plays an important role in matrix theory and its broad applications. In fact, certain kinds of characteristics of systems, such as solvability of linear equations and controUability/observability of linear control systems, are confirmed by evaluating the rank of specific matrices. However, the ordinary rank may not be sufficient for the precise description of the characteristics because of its integral value. As an example, let us consider two linear time-invariant systems, of which the state space equation is defined by dx - - = A x + Bu dt

(1)

where x and u are respectively state and control variables. Systems I and II are characterized by

System I:

System H:

A=

,

B=

.

(lb)

* Present address: Dept. of Planning and Management Science, Nagaoka University of Technology, Nagaoka 940-21, Japan. 0165-0114/90/$03.50 ( ~ 1990~Elsevier Science Publishers B.V. (North-Holland)

S. Kaguei, A. Ohsato

356

These two systems are state controllable [2] since both their controllability matrices, [B::AB], have rank 2. Then, which system is more controllable? This question is meaningless if A and B are precisely determined without errors. However, there may exist some ambiguity, such as disagreement between the real system and the model, errors in the parameter estimation and so on, which causes ambiguity of A and B. Therefore, it is possible that the real system may lose control even if controllability is assured from the rank of the matrix. In other words, the conventional rank of a matrix cannot indicate the grade of controllability. This seems to be caused by the integral value of the rank. As far as the authors know, the rank of matrix has not yet been valued in real numbers. Our obje6tive is to introduce a new type of real-valued rank which reflects the grade of some characteristics. In this paper, a new concept of rank is proposed for fuzzy matrices whose elements are fuzzy numbers.

2. F u z z y matrix a n d r a n k

2.1. Terminology A fuzzy number is usually defined as a convex, normalized fuzzy set on the real line whose membership function is piecewise continuous [4]. To avoid troublesome arguments, we impose some restrictions on the membership function as follows. Definition 1. A proper fuzzy number d is a normalized fuzzy set on the real line whose membership function f~(x) is continuous and has continuous inverse function on the support of x < a and x > a, where a is the mean value of ~i, i.e., f~(a) = 1. And, a fuzzy number is a crisp n u m b e r or a proper fuzzy number. Since the term 'fuzzy matrix' indicates in some cases a matrix over a semiring [3], we confirm its meaning (e.g. [1]). Definition 2. A fuzzy matrix is a rectangular array of fuzzy numbers. Its mean matrix is the crisp matrix composed by its mean values. 2.2. The function g(A) First of all, let us consider the square fuzzy matrix ,A of order k,

~/, =

i

!

.

(2)

akk_J

Ltlkl

Then, we introduce the following real-valued function of a square fuzzy matrix: g(A,) =

sup min{f~,(xll),f~12(x12), . . . ,f~a(Xkk)}

det X = 0

(3)

The rank of a

fuzzy

matrix

357

where fa,j(x) is the membership function of ~iij and X is a crisp matrix of order k, X=

"

"

kXkl

.

(3a)

X, k k d

The minimum value of the membership functions in (3) may be called the grade of equality between the singular crisp matrix X and the fuzzy matrix .,~. Therefore, g(.A) indicates the grade of singularity of A. Note that for crisp elements we can strike out its membership functions from (3) and replace the corresponding elements of X by its mean values. Also, it should be noted that the supremum in (3) is a maximum under the restriction of the membership functions of proper fuzzy numbers. The following properties are obtained from the definition.

Property 1. 0 ~
2.3. The rank of a fuzzy matrix Now, we may introduce the concept of rank for any m x n fuzzy matrix .A. Let us consider a minor of A of order k, which is obtained by striking out all but rows i l , . . . , i k and columns jl . . . . . J k from matrix ,~, written as •

ailh

:i:t

•..

ailjk

(4) t..aikh

aikjk ..J

Definition 3. Let k be the maximum order of a minor of A whose mean matrix is not singular, that is, det{A(jl:::~{)}=~0. The rank of a fuzzy matrix, r(.A), called margin rank in this paper, is defined as

r(t~)=k-

min gIA(i.l" " "ik~l il-..i k

I.

\ j 1 /

I~:::::~)[k- g{/~(ill :i]:)}].

= max

(5)

J

(5a)

It should be noted that the second term on the right hand side of (5) represents the margin of retaining the rank of the mean matrix in terms of membership values. When the matrix rank r(A) tends to the rank of the mean matrix r(A), the margin increases. If r ( A ) = r(A), the rank of matrix is not influenced by ambiguity of the matrix elements. The following properties are obtained directly from the definition and Properties 1-3.

S. Kaguei, A. Ohsato

358

Properly 4. If.4 is crisp then r(~t) coincides with the conventional rank for crisp

matrices. Property 5. r(A) - 1 < r(,4) <<-r(A). 2.4. Some examples Example 1. Consider the following fuzzy matrix:

where the membership functions of i and 2 are given in Figure l(a). Then g(A1) is 0.5 when X=

[1.51.5] 1.5 1 . 5 "

Therefore the margin rank of the fuzzy matrix ~i,1, r(~i.1), is 2 - 0.5 = 1.5. Example 2. Let

where [ and 2 are the fuzzy numbers in Example 1. Then g(A2) = 0.8 when X=

1.2

0.8 "

Therefore r(A2) = 2 - 0.8 = 1.2. These examples suggest that, in the example of Section 1, if the ambiguity of the elements of [B lAB] is represented by the fuzzy numbers 1 and 2 in Figure l(a), then System I is more controllable than System II. Example 3. Consider

where 2 is a crisp number and [ is the fuzzy number shown in Figure l(a). Then g(A3) = 3 - V~ ~ 0.76 when

Therefore r(.4a) ~ 1.24. Example 4. Let

where i and 2 are given by Figure l(b). Then g(A4) = 0. Therefore r(A4) = 2.

The rank of a fuzzy matrix

359

f~(x) O

0.5

1

2

3

f.(x) a m

0.5

o

o

0.8 1 1.2

1.8

3

Fig. 1. Membershipfunctionsof fuzzy numbers 1 and 2. (a) Examples 1-3; (b) Example 4. 3. Evaluation of rank The main part in calculating the rank of a fuzzy matrix is the evaluation of g(/t). In fact, it is very easy to obtain the rank from g(,g,) according to (5). A method of evaluating g(A) is discussed in this section. 3.1. Calculation of g ( A ) Similar to Section 2.2 we consider a square fuzzy matrix ~/, of order k. In the following arguments, it is assumed that the determinant of its mean matrix, d e t A , is positive. If d e t A < 0 then we may change the sign of a row or a column of ~/,. First of all, we will prove a lemma. Lemma. Assume that several elements of a square matrix, x#, lie between two real numbers x V*and x#**, that is, xij*
360

S. Kaguei, A. Ohsato

or d e t X < d e t X " ) , corresponding to the replaced elements vanish. such that d e t X ' < d e t X

all cofactors of X, X '

and X"

Proof. Let X* and X** be the matrices obtained by replacing the (i,j)-th element of X by x~ and xi~.*, respectively. Then d e t X - d e t X * = (x 0 - x ~ ) X o and detX** - d e t X = (x~* - x l j ) X o where X 0 is the cofactor of X with regard to the (i, j)-th element. Therefore, if X o = 0 then d e t X * = d e t X = det X**, if X 0 > 0 then d e t X * < d e t X < det X**, if X 0-< 0 then d e t X * > d e t X > det X**. Applying this process to other elements, the first part of this lemma is proved. If det X = det X ' then all the cofactors of the matrices corresponding to the replaced elements should vanish, since we can repeat the above process in any order. So the proof is completed. Let gO be a real n u m b e r between 0 and 1. Then the roots of fao(x) = go, x~ and x~*, can be obtained for all proper fuzzy numbers ~i0. Let X* be the matrices given by replacing all (i, j) elements of A by either x 0* or x o** . (The number of possible X* is 2 M, where M is the number of proper fuzzy elements.) We define the function F(go) as F(go) = min d e t X * .

(6)

X*

Then we may state the following propositions for F(go). Proposition 1. F(go) is continuous and non-decreasing with respect to go. I f the

matrix with the minimum determinant has a non-zero cofactor with regard to a proper f u z z y element then F( go) is increasing at this point. Proof. Since xi~ and x 0 are continuous functions of go, d e t X * is continuous. Therefore F(go) is also continuous. Let X ° be one of the matrices whose determinant is the minimum value, i.e., F ( g o ) = d e t X °. When go'< go, there exist such that fao(Xi'* Xi'* j and Xi '** j j ) = fao(Xij'** ) = go, and xij,* < x o0< x o,** . According to the lemma, F(go')~< F(go). Therefore, F(go) is a non-decreasing function. Also according to the lemma, F(go') < F(go) for go' < go if X ° has a non-zero cofactor. So the proof is completed. Proposition 2. Let goo be the maximum root o f F( go) = O. Then goo = g(2{ ). Proof. Since the demonstration that g(A) 1> goo is trivial, we will prove only that

g(A) -< go0. From the continuity offa,j(x) there exits a matrix X ° and a (p, q) element o f X ° such that det X ° = 0 and

g ( A ) = min{fa,,(x°,) . . . . .

fa~(X°kk) }

= ~q(XOpq) .

The rank of a fuzzy matrix

361

If there is an (i, j) element such that g(,4)
fa~(Xkk) } = g(,A ) for X* and detX* ~<0, then we obtain that F{g(A)} ~<0. Therefore g(,£,)~< 90So the proof is completed. min{f~H(Xn) . . . . .

From Propositions 1 and 2 we obtain the following sufficient condition for evaluating g(A).

Proposition 3. / f d e t X * / > 0 for all X* and there exists X ° in X* such that det X ° = 0 and X ° has a non-zero cofactor with regard to a proper f u z z y element, then g(A ) = ~p. Besides, the following proposition holds for the minimum matrix, X °.

Proposition 4. Consider the set of the above-mentioned matrices X*, i.e., their (i, j) element is either of the roots of fa,j(x) = dp when 5q is a proper f u z z y number. Let X ° be a matrix whose determinant is minimum. Then 0

0

X # ( X i j -- air) <~0

for all (i, j)

(7)

where X ° is a cofactor of X ° with regard to its (i, j) element, x °, and aq is the mean value of aij. Proof. Let x~ and x~* be the roots offn,j(x) = q~ such that x#* < x q** . Assume that X ° > 0 and x ° - x 0** for a certain (i, j) element. Let X' be the matrix such that the (i, j) element of X ° is replaced by x~. Then detX °-detX'-

__ ( X i j**

* 0 > 0, - x/~)X~

which contradicts the minimality of the determinant of X °. Similarly the same 0 * results hold true in case X~j < 0 and x ° = x 0. So the proof is completed. 3.2. Procedure for calculation of g ( A ) In most cases the membership function for a proper fuzzy number has a non-zero derivative on its support except in a finite number of points. Let us consider the procedure to solve the equation F ( t p ) = 0 under this assumption. Since F(tp) is piecewise differentiable, Newton's method may be used to solve the equation. The n-th approximate solution, tp,, is obtained from the (n - 1)-th one, ¢~n--1~ as

q,.= q,.-1 - F(¢.-1) / ( ~ ) ) ,

=,._~

(8)

where F(q~) = d e t X °,

(8a)

de(C) = X X °. / dq~

i.j

q/\

(8b) dx

ix=x,]"

S. Kaguei, A. Ohsato

362

Note that each term on the right hand side of (8b) is non-negative according to Proposition 4. In the neighborhood of q~ = 1, we obtain d e t X ~ d e t A + '~ Aij(xij - aq) 1,1

where Aq is a cofactor of A. If we choose xq such that Aq(xq - aq) <~0 for all i and j, d e t X is minimum. Therefore, when we start at tp = 1 to solve F(~p), the positive left derivatives have to be chosen for Aq > 0 and the negative right derivatives for Aq < 0 . As a matter of course, if all the derivatives vanish at = 1, we have to start from a point tp < 1. 3.3. Examples Example 5. Consider the linear mapping, y =.~x, where

A=

Ii i1 ~ -i

-i

-

0

We have to mind the following. We assume that the mapping itself is crisp but we do not have sufficient knowledge about the mapping. The ambiguity is assumed to be expressed by fuzzy numbers. We assume that the membership function of the fuzzy number ti is given as

f~(x)={~-lx-a]

f o r l x - a l ~<1, for Ix - a l > 1.

As an example, I), is shown in Figure 2. Then g ( . 4 ) ~ 0 . 6 9 when 3.69

5.31

0.31

5.69q

-031

069

O lj /

1.69 2.31 0.69-0.69

-0.69 -2.31[ 0.31-0.311 f-(x) o

I

-i

0

1

x

Fig. 2. Membershipfunction for fuzzy number 6 in Examples 5 and 6.

363

The rank of a fuzzy matrix

Table 1. Approximate solutions by Newton's method for Example 5 n

9.

F(~n)

(dF(~)/d~)~=,.

0 1 2

1 0.755 0.689

54.0 10.1 -0.1

220 154 174

and r(J,)~3.31. Therefore, the mapping is one-to-one when all membership values are greater than 0.69. Table 1 shows the approximate solutions using Newton's method. Since F(~)) is nearly linear, as shown in Figure 3, a rapid convergence is observed.

Example 6. Consider the linear equation ~ =/~, where A,=

and

/;=

.

Since the ranks of A and [A i b] are respectively 2 and 3, the equation Ax = b is not solvable. However, with the membership functions defined in Example 6, r([.4 I/~])~ 2.09 when X° ~

1.09 1.91 -1.09

1.09 0.91] -0:09 0.09 . 2.91 2.09

In this case, let X ° = [A' ! b']. Then the rank of A' is equal to that of [A' i b']. Therefore, there exists a solution when membership values are equal to 0.91. 60

50

40

30

20 10

0

0.6

i

0.7

0.8

0.9

Fig. 3. F(•) vs. ¢~for Example 5.

1

364

S. Kaguei, A. Ohsato

4. Concluding remarks We have introduced a new type of rank, called margin rank, for matrices with fuzzy elements. The value of this new rank is a real number that represents the margin in terms of grade of retaining the rank of the mean matrix. Several examples show that the margin rank enables us to assign an index that reflects the grading of some characteristics of the systems. In this paper the existence of inverse functions and their continuity are imposed on the membership functions of fuzzy numbers, but this may not be any obstacle to practical applications. Moreover, the arguments may be extended to general fuzzy numbers with little modifications. The calculation method employed here is not satisfactory in the sense that the calculation of 2 M determinants in (6), where M is the number of proper fuzzy elements, is not avoided. A more efficient algorithm is needed for fuzzy matrices with order greater than 4. The authors are presently working on the development of a more efficient algorithm to estimate the margin rank.

References [I] D. Dubois and H. Prade, Fuzzy Sets and Systems - Theory and Applications (Academic Press, New York, 1980). [2] P.L. Falb and M. Athans, A direct constructive proof of the criterion for complete controllability of time-invariant linear systems, 1EEE Trans. Automatic Control 9 (1964) 189-190. [3] K.H. Kim and F.W. Roush, Generalized fuzzy matrices, Fuzzy Sets and Systems 4 (1980) 293-315. [4] H.-J. Zimmermann, Fuzzy Set Theory- And Its Applications (Kluwer-Nijhoff, DordrechtBoston, 1985).