Applied Mathematical Sciences, Vol. 5, 2011, no. 6, 273 - 278

A New Criteria for a is not Generalized Strictly Diagonally Dominant Matrix

Zhijun Guo 1 and Jianguang Yang

Department of Mathematics, Hunan City University Yiyang, Hunan 413000, P.R. China

Abstract In this paper, we obtained some new results about a matrix is not generalized strictly diagonally dominant matrix by using gauss elimina- tion and some exists theories, which improve the corresponding results of [4]. Advantages are illustrated by numerical examples.

Mathematics Subject Classification: 15A47

Keywords: Gauss elimination, diagonally dominant matrix, diagonally dominant matrix, bi-diagonally dominant matrix

1 Introduction

Generalized strictly diagonally dominant matrix play an important role in numerical algebra, control theory, electric system, economic mathematics and elastic dynamics and so on. But it isn’t easy to determine whether a matrix is or not a generalized strictly diagonally dominant matrix in reality. In this paper, we obtained some new results about a matrix is not generalized strictly diagonally dominant matrix.

2 Preliminary Notes

n×n n×n n×n Let C (R ) be the set of complex (real) n×n matrices, A =(aij) ∈ C , N = {1, 2, ···,n}, for any i, j ∈ N, we denote: αi = |aij|,βi = |aij|, α˜i = |aji|, β˜i = |aji|, j∈N1 j∈N2 j∈N1 j∈N2 j=i j=i j=i j=i Λi = |aij| = αi + βi,Si = |aji| =˜αi + β˜i, j=i j=i [email protected] 274 Z. Guo and J. Yang

where N = N1 ⊕ N2,N1 = {i||aii| > Λi,i∈ N},N2 = {i|0 < |aii|≤Λi(A),i∈ (k) (k) N}. In this paper, we denote A =(aij ) is a matrix that A after kth steps gauss elimination. n×n Definition 2.1 Let A =(aij) ∈ C . (i) If |aii|≥Λi, ∀i ∈ N, then A is a diagonally dominant matrix; (ii) If |aii| > Λi, ∀i ∈ N, then A is a strictly diagonally dominant matrix, we denote it by A ∈ D0; (iii) If there exists a positive D = diag{d1,d2,...,dn}, such that AD = B ∈ D0, then A is a generalized strictly diagonally dominant matrix, we denote it by A ∈ D. It is obvious that if N1 = Φ be an empty set, then A ∈ D ,ifN2 = Φ, then A ∈ D. n×n Definition 2.2 Let A =(aij) ∈ C . (i) If there exists an α ∈ (0, 1] and

|aii|≥αΛi +(1− α)Si (1.1) for any i ∈ N, then A is an α diagonally dominant matrix. We denote it by A ∈ D0(α); (ii) If all strictly inequality holds in (1.1), then A is a strictly α diagonally dominant matrix. We denote it by A ∈ D(α); (iii) If there exists a positive diagonal matrix D, such that AD = B ∈ D(α), then A is a generalized strictly α diagonally dominant matrix. We denote it by A ∈ D(α). n×n Definition 2.3 Let A =(aij) ∈ C . (i) If there exists an α ∈ (0, 1] and α 1−α |aiiajj|≥(ΛiΛj) (SiSj) (1.2) for any i = j, (i, j ∈ N), then A is an α bi-diagonally dominant matrix. We α denote it by A ∈ D0 ; (ii) If all strictly inequality holds in (1.2), then A is a strictly α bi-diagonally dominant matrix. We denote it by A ∈ Dα; (iii) If there exists a positive diagonal matrix D, such that AD = B ∈ Dα, then A is a generalized strictly α bi-diagonally dominant matrix. We denote it by A ∈ Dα. n×n n×n Definition 2.4 Let A =(aij) ∈ C ,ifμ(A)=(mij) ∈ R , |aii|,i= j, mij = −|aij|,i= j. then we say μ(A) is a comparison matrix of A. n×n A matrix A =(aij) ∈ C is called nonsingular H-matrix if its comparison matrix μ(A) is a nonsingular M-matrix. Criteria for a matrix is not generalized strictly diagonally dominant matrix 275

3 Main Results

In this section, we will give some criteria for a matrix is not generalized strictly diagonally dominant matrix.

Lemma 3.1 ([1]) If A ∈ D(α),then μ(A) is a nonsingular M-matrix and A ∈ D.

Lemma 3.2 ([2]) If A ∈ Dα,then A ∈ D.

n×n Lemma 3.3 ([3]) Let A =(aij) ∈ C ,ifA is a nonsingular H-matrix, then there exists at least one strict diagonally dominant row, namely N1 =Φ .

n×n Lemma 3.4 ([4]) Let A =(aij) ∈ C , if there exist N1, N2 such that N1 ∪ N2 = N, and (|aii|−αi)(|ajj|−βj) ≤ βiαj for any i ∈ N1,j ∈ N2, then A is not a generalized strictly diagonally dominant matrix and μ(A) is not a nonsingular M-matrix.

n×n (n−1) Theorem 3.5 Let A =(aij) ∈ C ,ifA ∈ D0, then A ∈ D0.

Proof. As A ∈ D0, we have a11 = 0 and |aii| > |aij| = |ai1| + |aij| j=i j=1,i Λ1 > |ai1| + |aij| |a11| j=1,i |ai1| |ai1| = |a1j| + |a1i| + |aij| |a11| j=1,i |a11| j=1,i it follows that

|ai1| |ai1| |aii|− |a1i| > |aij| + |a1j| |a11| j=1,i j=1,i |a11| for matrix A(1). (i) When i = 1. It is obvious that

(1) (1) |a11 | = |a11| > |a1j| = |a1j | j=1 j=1 276 Z. Guo and J. Yang

(ii) When 1

(1) a1iai1 |ai1| |aii | = |aii − | > |aii|− |a1i| a11 |a11| |ai1| > |aij| + |a1j| j=1,i j=1,i |a11| ai1a1j > |aij − | j=1,i a11 (1) (1) = |aij | =Λi(A ) j=1,i (1) thus A ∈ D0, in the same way, after n − 1 steps gauss elimination, we have (n−1) A ∈ D0. n×n (n−1) Theorem 3.6 Let A =(aij) ∈ C ,ifA ∈ D , then A ∈ D . Proof. As A ∈ D, there exists a positive diagonal matrix D satisfied Δ AD = B ∈ D0. We construct the inverse Frobenius matrix as follow: ⎛ ⎞ 1 000 ⎜ ⎟ ⎜ − ⎟ −1 ⎜ c21 100⎟ L 1 = ⎜ . . . . ⎟ ⎝ . . .. . ⎠ −cn1 ··· ··· 1

ai1 −1 −1 −1 i1 ··· where c = a11 , (i =2, 3, ,n), since L1 B = L1 (AD)=(L1 A)D = (1) (1) (1) A D, by Theorem 3.1 we know A D ∈ D0, therefore A ∈ D , in the same way, we have A(n−1) ∈ D. n×n (n−1) Theorem 3.7 Let A =(aij) ∈ C ,ifA ∈ D (α), then A ∈ D . Proof. By definition 2.2 and Lemma 3.1 we know that there exists a positive diagonal matrix D1 satisfied AD1 ∈ D , furthermore, there exists a Δ Δ positive diagonal matrix D2 such that AD1D2 = AD = B ∈ D0, by theorem 3.2, we have A(n−1) ∈ D. n×n α (n−1) Theorem 3.8 Let A =(aij) ∈ C ,ifA ∈ D , then A ∈ D . Proof. By Lemma 3.2 and Theorem 3.2 we can obtain the result. n×n α (n−1) Theorem 3.9 Let A =(aij) ∈ C ,if A ∈ D , then A ∈ D . Proof. By definition 1.3 we know that there exists a positive diagonal α matrix D1 satisfied AD1 ∈ D , then by Lemma 2.2, AD1 ∈ D , furthermore, Δ Δ there exists a positive diagonal matrix D2 such that AD1D2 = AD = B ∈ D0, by Theorem 3.2, we have A(n−1) ∈ D. n×n (n−1) Theorem 3.10 Let A =(aij) ∈ C ,ifA ∈ D , then A ∈ D . Proof. Suppose A ∈ D, by Theorem 3.2 we have A(n−1) ∈ D, the results conflicts with the conditions of this theorem, thus A ∈ D. Criteria for a matrix is not generalized strictly diagonally dominant matrix 277

4 Numercial example

In this section, we give some examples to illustrate our results is effective. Example 4.1 Let ⎛ ⎞ 10.50.4 ⎜ ⎟ A = ⎝ 0.310.5 ⎠ 0.10.61 obviously A ∈ D0, since ⎛ ⎞ 10.50.4 ⎜ ⎟ A(2) = ⎝ 00.85 0.38 ⎠ 607 00 850 (2) so A ∈ D0. Example 4.2 Let ⎛ ⎞ 10.40.20.1 ⎜ ⎟ ⎜ 0.310.10.2 ⎟ A = ⎜ ⎟ ⎝ 00.510.4 ⎠ 0.80.40.51 1 1 1 ∀ ∈{ }  | |≥ 2 2 when α = 2 , i, j 1, 2, 3, 4 , (i = j), we have aiiajj (ΛiΛj) (SiSj) ,by definition 2.3, A ∈ Dα, since ⎛ ⎞ 10.40.20.1 ⎜ ⎟ (3) ⎜ 00.88 0.04 0.17 ⎟ A = ⎜ ⎟ ⎝ 000.9773 0.3034 ⎠ 00 0 0.8001 (3) so A ∈ D0. Example 4.3 Let ⎛ ⎞ 10.11 ⎜ ⎟ A = ⎝ 011.4 ⎠ 11.12.4 since (|a33|−α3)(|a11|−β1)=2.16 > 2.1=β3α1 so A does not satisfy the conditions of the Theorem 3 in [4], that is the Lemma 3.4 in this paper, but ⎛ ⎞ 10.11 ⎜ ⎟ A(2) = ⎝ 011.4 ⎠ 00 0 so A does not satisfy Lemma 3.3, A(2) ∈ D, by Theorem 3.6, we have A ∈ D.

Acknowledgements. Projects supported by the Science and Technology Foundation of Yiyang, Hunan, China(Grant No. 2010JZ20) and the Research Foundation of Education Bureau of Hunan Province(Grant No. 10B015). 278 Z. Guo and J. Yang

References

[1] Y. X. Sun. Sufficient conditions for generalized diagonally dominant ma- trices. Numerical Mathematics A Journal of Chinese Universities. 3(1997), 216-223.

[2] Q. C. Li. Determinate conditions of generalized strictly diagonally domi- nant matrices. Numerical Mathematics A Journal of Chinese Universities. 1(1999), 87-92.

[3] J. C. Li. W. X. Zhang. Criteria for H-matrices. Numerical Mathematics A Journal of Chinese Universities. 3(1999), 264-268.

[4] Y. M. Gao, X. H. Wang, Criteria for Generalized Diagonally Dominant Matrices and M-matrices, Appl. 169(1992), 257-268.

[5] Y. M. Gao, X. H. Wang, Criteria for Generalized Diagonally Dominant Matrices and M-matrices.II, Linear Algebra Appl. 248(1996), 339-353.

[6] R. J. Plemmons, A. Bermon, Nonnegative Matrices in the Mathematical Sciences, SIAM press, Philadelphia, 1994.

[7] L. Li, On the iterative criterion for generalized diagonally dominant ma- trices. SIAM J. Matrix Anal. Appl. 24(2002), 17-24.

[8] T. B. Gan, T. Z. Huang. Simple Criteria for Nonsingular H-matrices. Linear Algebra Appl. 374(2003), 217-326.

[9] T. B. Gan, T. Z. Huang. Practical Sufficient Conditions for Nonsingular H-matrices. Mathematica Numerica Sinica, 26(2004), 109-116.

Received: August, 2010