Научни трудове на Съюза на учените в България-Пловдив, серия Б. Естествени и хуманитарни науки, т.ХУЬ Научна сесия „Техника и технологии, естествени и хуманитарни науки", 30-31 Х 2013 Scientific researches of the Union of Scientists in Bulgaria-Plovdiv, series B. Natural Sciences and the Humanities, Vol. XVI.,ISSN 1311-9192, Technics, Technologies, Natural Sciences and Humanities Session, 30-31 October 2013

PROPERTIES AND APPLICATION OF M-MATRICES

Mariyan Milev, University of Food Technology - Plovdiv, Bulgaria marianmilev2002@gmail.com Mariyan Iliev, Technical University - Sofia, Bulgaria iliev_m@abv.bg

Abstract: In this article we explore definitions and structure of M-matrices. In order to define the general structure of M-matrices we have presented theorems and properties of some nonnegative and irreducible matrices. We explore tridiagonal matrices as a special class of M-matrices that is most frequently used in applied mathematics.

1. Introduction.

The necessity of studying M-matrices has emerged from the numerous application of matrices with such structure in numerical analysis. It turns out that such kind of matrices have a variety of useful properties that makes them a valuable tool in Biomatematics, Robotics and Finance [1], [4], [5]. For example, the nonnegativity of the inverse of a M-matrix, because most models describing real life and financial markets utilize nonnegative variables.

Another advantage of M-matrices is their application when solving parabolic partial differential equations such as the Black-Scholes one [3]. New or modified finite differences schemes could be invented using M-matrices because the stability of traditional finite differences schemes fails in cases of partial differential equations with non-smooth boundary conditions [6]. A confirmation of our words is the book of Samuli Ikonen [4]. He shows that this class of M-matrices is often a sufficient condition to be obtained a numerical solution that does not oscillate.

All these arguments encourage us to present a study devoted on the structure and main properties of M-matrice that is not a central topic in traditional books for matrix theory and analysis such as [2], [7]. A lot of information for M-matrices is proposed by Windish in [11], Varga in [9]. A short review for M-matrices is presented in [8], [10].

In the next section we present different definitions of M-matrices that are frequently used in practice. We present important theorems that concern the structure of a given M-matrix and its inverse one. In Section 3 we present some special examples of M-matrices, i.e. tridiagonal matrix. In Section 4 we explore the sum and product of two M-matrices. In the conclusion we give some final remarks for the wide application of M-matrices.

2. Definition and Structure of M-matrices

Definition 2.1: All matrices are real. A matrix A = (aij-) is called an M-matrices if aij < 0 whenever i Ф j and all principal minors of A are positive.

Definition 2.2: (Fan). A matrix A with nonpositive off-diagonal elements is an M-matrix if and only if A is nonsingular and A 1 is nonnegative.

Definition 2.3: Any matrix A of the form A = sI — B , with s > 0, B > 0 for which s > S(B), where S(B) is the spectral radius of the nonnegative matrix B , is called an M-matrix. For s = S(B) , the matrix A is a singular M-matrix.

Obviously, aij < 0 whenever i * j . For s > S(B) , any M-matrix is a monotone

matrix (A matrix A for which det A * 0 and A 1 > 0 ), but conversely is not true.

The three definitions for M-matrices are equivalent for the nonsingular case det A * 0 .

In literature there are two fundamental approaches for exploring M-matrices.

1. If N denotes the class of nonsingular nonnegative matrices, determine those matrices A in N for which A 1 is an M-matrix.

2. Among the class LLxn of matrices A = (aij-) satisfying aii > 0 for each i and

a„ < 0 whenever i * j , determine those for which A 1 > 0 .

v

We present one fundamental theorem and properties that describes the general structure of M-matrices.

Theorem 2.1: Any nonsingular M-matrix A = (a^ ) and its inverse A 1 = (ciij ) have all positive diagonal entries.

Proof: From the identity A A'1 = lit follows that: ^ aik(Xlci = 1 , Vi e N

keN

By ajj < 0, i * j and A1 > 0, we have: aiieiii = 1 — ^aikeilci > 1

k

Thus, ¿tii > 0 instead of 0lii > 0 , and additionally au > 0 , Vi e N . □

In order to define conditions for M-matrices let us denote: Z- ={ A = a): atj < 0, i * j} and

Lnxn = {A = (av): atj < 0, i * j and au > 0, Vie N}. Obviously: Lnxn c Znxn.

From the definition of M-matrices, the class of M-matrices is a subset of Z"x" . From Theorem 1, follows that the class of nonsingular M-matrices is a subset of Lnxn .

We present the most famous theorems for M-matrices from the classes Z"x" and Lxn . Theorem 2.2: A matrix A e Z"xn is a nonsingular M-matrix if and only if the real parts of all eigenvalues of A are positive. (Matrices with Re X > 0 are called positive stable matrices.) Theorem 2.3: Let A e Lnxn which is strongly row or column diagonally dominant, i.e Ae > 0 or e A > 0 , where e = (1,...,1X . Then A is a nonsingular M-matrix.

Theorem 2.4: Let A be an irreducible L-matrix which is weakly row or column diagonally dominant, i.e. Ae > 0, * 0 or A > 0T, * 0 . Then A is a nonsingular M-matrix.

We remember that an n x n real or complex matrix A = (a.•) is diagonally dominant if

\ij\' i = n ■ The matrix A = (a.) is strictly diagonally dominant if strict

j*i

inequalities hold in |aiJ |a.| for all i = l, — ., n ■ The matrix A = (a.) is irreducibly

j *i

diagonally dominant it is irreducible and if strict inequality holds in |aiJ > ^ \ai\ for at least

j *i

one i■ We remember that an n X n matrix A is reducible if there is a permutation matrix P such that:

PAP1 =

An A 0 A

-12

-22.

where A11 and A22 are square submatrices. The matrix S is irreducible if it is not reducible.

Obviously, any matrix all of whose elements are nonzero is irreducible. In particular, positive matrices are irreducible■ On the other hand, any matrix that has a zero row or column is reducible■

3. Some special examples of M-matrices: Tridiagonal M-matrices.

Let A = (aij) be tridiagonal, Le^ a. = 0 for | i — j | > 2 and has the following mode:

f c — b \

A =

a2 C2 b2

- an-1 Cn y

Let c ^ 0 , Vi G n . Let ai+1 > 0, b, > 0 for i = 1,..., n -1, which conditions

equivalent to the irreducibility of A = (a iJ- ) . The reducible case, i.e j j bi = 0 or j j ai = 0,

i=1 i=2

can be reduced to a sequence of irreducible tridiagonal submatrices.

Theorem 3.1: For the given above matrix A = (aiJ- ) there exists a positive diagonal matrix D = diag(d1,...,dn) such that A = D-1 AD is symmetric.

( „

The matrix D can be chosen that: d, = 1, d, =

1

^ 2

vbi ."bi-1 y

i = 2,

, n , where:

A = D- AD =

■■Jb

Jb~2

-1an cn y

= (a ' )

The matrices A' and A are similar, so they have the same spectrum. Thus, the spectrum of A is real, and additionally all eingenvalues of A are simple.

Theorem 3.2: An irreducible tridiagonal matrix A £ Z"/n is a nonsingular (singular) M-matrix if and only if the smallest eigenvalue of A is positive (nonnegative).

In some special cases of tridiagonal matrices A £ Zn/n we know the whole spectrum of eigenvalues. This is the case if A is a tridiagonal Toeplitz matrix, i.e.

A = tridiag (-a, c, -b) =

( c - b -a c - b

- a c

where a > 0, b > 0 . The spectrum of eigenvalues S(A) of the matrix A is given by:

S(A) = \^k = c -2yfab cos

k n n +1

k £ N

According to Theorem 3.2 it follows that the matrix A is a nonsingular M-matrix if

c - 2yj ab > 0. For b = a and c - 2 a > 0, the matrix A is a Stieltjes matrix, i.e. a symmetric nonsingularM-matrices. We remember that any Stieltjes matrix is positive definite.

Then the corresponding system of eigenvectors is:

k n

i-.

n +1

sin 2

k n n +1

sinn

k n n +1

, k £ N

Thus using Theorem 3.2 we prove analytically that the symmetric irreducible matrix A = tridiag(-1, 2,-1) , A £ Zn/n , is a nonsingular M-matrix.

A =

2 -1 -1 '•.

-1

-1 2

We can directly prove this using Theorem 2.4 because A is an irreducible L-matrix which is weakly row diagonally dominant. In particular, it is true that: A 1 > 0 . The last matrix A = tridiag(-1, 2,-1) is fundamental when the explicit Euler finite difference scheme is applied for numerical solution of the heat equation.

Finally, we can conclude the following theorem for the inverse of tridiagonal matrix with positive diagonal elements and negative off-diagonal elements.

T

Theorem 3.3: Let A = (a j) is a tridiagonal with positive diagonal elements and negative offdiagonal elements. If the matrix A = (a j) is strictly or irreducibly diagonally dominant then A = (aij ) is a nonsingular M-matrix and A 1 > 0 .

4. Properties of M-matrices: Sum and Product of M-matrices

The class of monotone matrix is closed under matrix multiplication. In general, the sum and the product of two M-matrices is not an M-matrix.

f 10^ T Examples: Let: A1 = and A 2 = A1 , where both matrices are nonsingular M-matrices

I- a 1J

for any a > 0. Then the sum A1 + A 2 is an M-matrix only for 0 < a < 2 (the sum is nonsingular for 0 < a < 2 and a singular M-matrix for a = 2).

For the matrix A = tridiag( — a,1,0) of order n = 3 , which is a nonsingular M-matrix for any a > 0 , A 2 £ Z 3x3 holds only for a = 0 . Therefore, A 2 is an M-matrix only for a=0.

By special assumptions, the sum and the product of two M-matrices may be an M-matrix.

Property 1: Let A1 and A 2 be M-matrices which are strongly row diagonally dominant, that is, Ai e > 0 , i = 1,2 and e = (1,.. .,1)T . The sum A1 + A2 is an M-matrix, which is also strongly row diagonally dominant, because (A1 + A2)e = A1 e + A2 e > 0 .

Property 2: Let A1 and A 2 be M-matrices where A1 A 2 £ Z"/n . Then the product A1 A 2 is an M-matrix.

The class of nonsingular M-matrix is closed under positive diagonal multiplication. Property 3: If A = (a^) is a nonsingular M-matrix and D is a positive diagonal matrix,

then AD and DA are nonsingular M-matrix.

Property 4: The product of two nonsingular M-matrices is in any case a monotone matrix. Property 5: The 2 / 2 — M — matrices are closed under matrix multiplication.

Proof: Let A i =

f a — b ^

V— ci di V

i = 1,2 . The we have A1 A2 £ Zn/" by:

AA =f a1 a2 + b1b2 — (a1b2 + b1d 2 ) ^ (c1a2 + d1c2) c1c2 + d1d2 j Property 6: For any nonsingular M-matrix A = (a^) there exists a vector x > 0 such that Ax > 0.

1. It is known that if A > B are real square matrices with nonnegative inverse, then

A - < B

2. If 0 < A < B , then p(A) < p(B) .

Property 7: Let A = (a.) be an M-matrix and D a nonnegative diagonal matrix. Then A + D is an M-matrix, and (A + D) -1 < A "V

The class ofM-matrices is closed under permutation cogredient operation.

Property 8: For any M-matrix A = (a.) and each permutation matrix P the matrix

A = P'AP is also an M-matrix.

5. Conclusions:

We have presented the structure of M-matrices that are a special class of matrices with a positive inverse matrix. Such matrices are particularly used in numerical methods for solving partial differential equations that are fundamental in mathematical models of real problems such as the heat transfer, transport and reaction of chemical species, population dynamics, robotics, assorption of pollutants in soil and diffusion of neutrons. All these quantities in real life may take only positive values and modern numerical methods use M-matrices to produce positive and smooth solutions without undesired spurious oscillations. The necessity of exploring properties of tridiagonal M-matrices emerges not only in physical and biological phenomena but also in quantitative models as the financial requirement of obtaining positive prices is irreplaceable.

6. References

1. B. M. Chen-Charpentier, H. V. Kojouharov: An Unconditionally Positivity Preserving Scheme for Advection-Diffusion Reaction Equations, Mathematical and Computer Modelling, 57 (2013), 2177-2185.

2. R. Horn, Ch. Jonson, Matrix Analysis, England, Cambidge University Press 1986

3. Samuli Ikonen, Efficient Numerical Solution of the Black-Scholes Equation by Finite Difference Method, PhD thesis, University of Jyvaskyla, Department of Mathematical Information Technology, Jyvaskyla, Finland, 2003.

4. N. Kochev, A. Terziski, Mariyan Milev, Numerical Modelling of Three-Phase Mass Transition with an Application in Atmospheric Chemistry, Applied Mathematics, Vol.4 No.8A, 100-106, 2013, DOI: 10.4236/am.2013.48A014, ISSN: 2152-7385 (Print), ISSN: 2152-7393 (Online) http://www.scirp.org/journal/AM/

5. Mariyan Milev, Aldo Tagliani, Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing, Serdica Mathematical Journal, Vol. 36 (n.1) (2010), 75-88, ISSN: 1310-6600, http://www.math.bas.bg/serdica/n1_10.html

6. Mariyan Milev, Aldo Tagliani, Efficient Implicit Scheme with Positivity-Preserving and Smoothing Properties, Journal of Computational and Applied Mathematics, Vol. 243 (2013), 1-9, ISSN: 0377-0427, http://dx.doi.org/10.1016/j.cam.2012.09.039, http://www.sciencedirect.com/science/article/pii/S0377042712004128

7. J. M. Ortega, Matrix Theory, Plenum Press, New York, 1988.

8. George Poole and Thomas Boullion, A Survey on M-matrices, SIAM, rev. 16 (1974), pp. 419-427.

9. S. Richard Varga, Matrix Iterative Analysis, Springer, Berlin, 2002

10. V. V. Voevodin, Ju. A. Kuznetzov, Matrices and Computations, Russia, Moscow 1984.

11. G. Windish, M-matrices in Numerical Analysis, Teubner-Texte zur Mathematik, 115, Leipzig, 1989.