Some inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse

  • Guanghui Cheng1Email author,

    Affiliated with

    • Qin Tan1 and

      Affiliated with

      • Zhuande Wang1

        Affiliated with

        Journal of Inequalities and Applications20132013:65

        DOI: 10.1186/1029-242X-2013-65

        Received: 31 July 2012

        Accepted: 24 January 2013

        Published: 21 February 2013

        Abstract

        In this paper, some new inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse are given. These inequalities are sharper than the well-known results. A simple example is shown.

        AMS Subject Classification:15A18, 15A42.

        Keywords

        Hadamard product M-matrix inverse M-matrix strictly diagonally dominant matrix eigenvalue

        1 Introduction

        A matrix A = ( a i j ) R n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq1_HTML.gif is called a nonnegative matrix if a i j 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq2_HTML.gif. A matrix A R n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq3_HTML.gif is called a nonsingular M-matrix [1] if there exist B 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq4_HTML.gif and s > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq5_HTML.gif such that
        A = s I n B and s > ρ ( B ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equa_HTML.gif
        where ρ ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq6_HTML.gif is a spectral radius of the nonnegative matrix B, I n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq7_HTML.gif is the n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq8_HTML.gif identity matrix. Denote by M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq9_HTML.gif the set of all n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq8_HTML.gif nonsingular M-matrices. The matrices in M n 1 : = { A 1 : A M n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq10_HTML.gif are called inverse M-matrices. Let us denote
        τ ( A ) = min { Re λ : λ σ ( A ) } , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equb_HTML.gif
        and σ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq11_HTML.gif denotes the spectrum of A. It is known that [2]
        τ ( A ) = 1 ρ ( A 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equc_HTML.gif
        is a positive real eigenvalue of A M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq12_HTML.gif and the corresponding eigenvector is nonnegative. Indeed
        τ ( A ) = s ρ ( B ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equd_HTML.gif

        if A = s I n B http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq13_HTML.gif, where s > ρ ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq14_HTML.gif, B 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq4_HTML.gif.

        For any two n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq8_HTML.gif matrices A = ( a i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq15_HTML.gif and B = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq16_HTML.gif, the Hadamard product of A and B is A B = ( a i j b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq17_HTML.gif. If A , B M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq18_HTML.gif, then A B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq19_HTML.gif is also an M-matrix [3].

        A matrix A is irreducible if there does not exist a permutation matrix P such that
        P A P T = [ A 1 , 1 A 1 , 2 0 A 2 , 2 ] , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Eque_HTML.gif

        where A 1 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq20_HTML.gif and A 2 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq21_HTML.gif are square matrices.

        For convenience, the set { 1 , 2 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq22_HTML.gif is denoted by N, where n (≥3) is any positive integer. Let A = ( a i j ) R n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq1_HTML.gif be a strictly diagonally dominant by row, denote
        http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equf_HTML.gif
        Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix have been proposed. Let A M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq23_HTML.gif, for example, τ ( A A 1 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq24_HTML.gif has been proven by Fiedler et al. in [4]. Subsequently, τ ( A A 1 ) > 1 n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq25_HTML.gif was given by Fiedler and Markham in [3], and they conjectured that τ ( A A 1 ) > 2 n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq26_HTML.gif. Song [5], Yong [6] and Chen [7] have independently proven this conjecture. In [8], Li et al. improved the conjecture τ ( A A 1 ) 2 n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq27_HTML.gif when A 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq28_HTML.gif is a doubly stochastic matrix and gave the following result:
        τ ( A A 1 ) min i { a i i s i R i 1 + j i s j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equg_HTML.gif
        In [9], Li et al. gave the following result:
        τ ( A A 1 ) min i { a i i m i R i 1 + j i m j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equh_HTML.gif
        Furthermore, if a 11 = a 22 = = a n n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq29_HTML.gif, they have obtained
        min i { a i i m i R i 1 + j i m j i } min i { a i i s i R i 1 + j i s j i } , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equi_HTML.gif

        i.e., under this condition, the bound of [9] is better than the one of [8].

        In this paper, our motives are to improve the lower bounds for the minimum eigenvalue τ ( A A 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq30_HTML.gif. The main ideas are based on the ones of [8] and [9].

        2 Some preliminaries and notations

        In this section, we give some notations and lemmas which mainly focus on some inequalities for the entries of the inverse M-matrix and the strictly diagonally dominant matrix.

        Lemma 2.1 [6]

        Let A R n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq3_HTML.gif be a strictly diagonally dominant matrix by row, i.e.,
        | a i i | > j i | a i j | , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equj_HTML.gif
        If A 1 = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq31_HTML.gif, then
        | b j i | k j | a j k | | a j j | | b i i | , j i , j N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equk_HTML.gif
        Lemma 2.2 Let A R n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq3_HTML.gif be a strictly diagonally dominant M-matrix by row. If A 1 = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq31_HTML.gif, then
        b j i | a j i | + k j , i | a j k | m k i a j j b i i u j b i i , j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equl_HTML.gif
        Proof Firstly, we consider A R n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq3_HTML.gif is a strictly diagonally dominant M-matrix by row. For i N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq32_HTML.gif, let
        r i ( ε ) = max j i { | a j i | + ε a j j k j , i | a j k | } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equm_HTML.gif
        and
        m j i ( ε ) = r i ( ε ) ( k j , i | a j k | + ε ) + | a j i | a j j , j i . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equn_HTML.gif
        Since A is strictly diagonally dominant, then r j i < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq33_HTML.gif and m j i < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq34_HTML.gif. Therefore, there exists ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq35_HTML.gif such that 0 < r i ( ε ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq36_HTML.gif and 0 < m j i ( ε ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq37_HTML.gif. Let us define one positive diagonal matrix
        M i ( ε ) = diag ( m 1 i ( ε ) , , m i 1 , i ( ε ) , 1 , m i + 1 , i ( ε ) , , m n i ( ε ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equo_HTML.gif
        Similarly to the proofs of Theorem 2.1 and Theorem 2.4 in [8], we can prove that the matrix A M i ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq38_HTML.gif is also a strictly diagonally dominant M-matrix by row for any i N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq32_HTML.gif. Furthermore, by Lemma 2.1, we can obtain the following result:
        m j i 1 ( ε ) b j i | a j i | + k j , i | a j k | m k i ( ε ) m j i ( ε ) a j j b i i , j i , j N , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equp_HTML.gif
        i.e.,
        b j i | a j i | + k j , i | a j k | m k i ( ε ) a j j b i i , j i , j N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equq_HTML.gif
        Let ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq39_HTML.gif to get
        b j i | a j i | + k j , i | a j k | m k i a j j b i i u j b i i , j i , j N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equr_HTML.gif

        This proof is completed. □

        Lemma 2.3 Let A = ( a i j ) M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq40_HTML.gif be a strictly diagonally dominant matrix by row and A 1 = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq41_HTML.gif, then we have
        1 a i i b i i 1 a i i j i | a i j | u j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equs_HTML.gif
        Proof Let B = A 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq42_HTML.gif. Since A is an M-matrix, then B 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq4_HTML.gif. By A B = B A = I n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq43_HTML.gif, we have
        1 = j = 1 n a i j b j i = a i i b i i j i | a i j | b j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equt_HTML.gif
        Hence
        1 a i i b i i , i N , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equu_HTML.gif
        or equivalently,
        1 a i i b i i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equv_HTML.gif
        Furthermore, by Lemma 2.2, we get
        1 = a i i b i i j i | a i j | b j i ( a i i j i | a i j | u j i ) b i i , i N , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equw_HTML.gif
        i.e.,
        b i i 1 a i i j i | a i j | u j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equx_HTML.gif

        Thus the proof is completed. □

        Lemma 2.4 [10]

        Let A C n × n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq44_HTML.gif and x 1 , x 2 , , x n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq45_HTML.gif be positive real numbers. Then all the eigenvalues of A lie in the region
        1 n { z C : | z a i i | x i j i 1 x j | a j i | } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equy_HTML.gif

        Lemma 2.5 [11]

        If A 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq28_HTML.gif is a doubly stochastic matrix, then A e = e http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq46_HTML.gif, A T e = e http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq47_HTML.gif, where e = ( 1 , 1 , , 1 ) T http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq48_HTML.gif.

        3 Main results

        In this section, we give two new lower bounds for τ ( A A 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq30_HTML.gif which improve the ones in [8] and [9].

        Lemma 3.1 If A M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq23_HTML.gif and A 1 = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq41_HTML.gif is a doubly stochastic matrix, then
        b i i 1 1 + j i u j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equz_HTML.gif

        Proof This proof is similar to the ones of Lemma 3.2 in [8] and Theorem 3.2 in [9]. □

        Theorem 3.1 Let A M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq23_HTML.gif and A 1 = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq41_HTML.gif be a doubly stochastic matrix. Then
        τ ( A A 1 ) min i { a i i u i R i 1 + j i u j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaa_HTML.gif
        Proof Firstly, we assume that A is irreducible. By Lemma 2.5, we have
        a i i = j i | a i j | + 1 = j i | a j i | + 1 and a i i > 1 , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equab_HTML.gif
        Denote
        u j = max i j { u j i } = max { | a j i | + k j , i | a j k | m k i a j j } , j N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equac_HTML.gif
        Since A is an irreducible matrix, we know that 0 < u j 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq49_HTML.gif. So, by Lemma 2.4, there exists i 0 N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq50_HTML.gif such that
        | λ a i 0 i 0 b i 0 i 0 | u i 0 j i 0 1 u j | a j i 0 b j i 0 | , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equad_HTML.gif
        or equivalently,
        | λ | a i 0 i 0 b i 0 i 0 u i 0 j i 0 1 u j | a j i 0 b j i 0 | a i 0 i 0 b i 0 i 0 u i 0 j i 0 1 u j | a j i 0 | u j b i 0 i 0 (by Lemma 2.2) ( a i 0 i 0 u i 0 j i 0 | a j i 0 | ) b i 0 i 0 = ( a i 0 i 0 u i 0 R i 0 ) b i 0 i 0 a i 0 i 0 u i 0 R i 0 1 + j i 0 u j i 0 (by Lemma 3.1) min i { a i i u i R i 1 + j i u j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equae_HTML.gif
        Secondly, if A is reducible, without loss of generality, we may assume that A has the following block upper triangular form:
        A = [ A 11 A 12 A 1 K 0 A 22 A 2 K 0 0 0 0 0 A K K ] , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaf_HTML.gif

        where A i i M n i http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq51_HTML.gif is an irreducible diagonal block matrix, i = 1 , 2 , , K http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq52_HTML.gif. Obviously, τ ( A A 1 ) = min i τ ( A i i A i i 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq53_HTML.gif. Thus the reducible case is converted into the irreducible case. This proof is completed. □

        Theorem 3.2 If A = ( a i j ) M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq40_HTML.gif is a strictly diagonally dominant by row, then
        min i { a i i u i R i 1 + j i u j i } min i { a i i s i R i 1 + j i s j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equag_HTML.gif
        Proof Since A is strictly diagonally dominant by row, for any j i http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq54_HTML.gif, we have
        d j m j i = | a j i | + k j , i | a j k | a j j | a j i | + k j , i | a j k | r i a j j = ( 1 r i ) k j , i | a j k | a j j 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equah_HTML.gif
        or equivalently,
        d j m j i , j i , j N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equ1_HTML.gif
        (1)
        So, we can obtain
        u j i = | a j i | + k j , i | a j k | m k i a j j | a j i | + k j , i | a j k | d k a j j = s j i , j i , j N , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equ2_HTML.gif
        (2)
        and
        u i s i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equai_HTML.gif
        Therefore, it is easy to obtain that
        a i i u i R i 1 + j i u j i a i i s i R i 1 + j i s j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaj_HTML.gif
        Obviously, we have the desired result
        min i { a i i u i R i 1 + j i u j i } min i { a i i s i R i 1 + j i s j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equak_HTML.gif

        This proof is completed. □

        Theorem 3.3 If A = ( a i j ) M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq40_HTML.gif is strictly diagonally dominant by row, then
        min i { a i i u i R i 1 + j i u j i } min i { a i i m i R i 1 + j i m j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equal_HTML.gif
        Proof Since A is strictly diagonally dominant by row, for any j i http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq54_HTML.gif, we have
        r i m j i = r i | a j i | + k j , i | a j k | r i a j j = r i | a j i | k j , i | a j k | a j j = r i ( a j j k j , i | a j k | ) | a j i | a j j = a j j k j , i | a j k | a j j ( r i | a j i | a j j k j , i | a j k | ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equam_HTML.gif
        i.e.,
        r i m j i , j i , j N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equ3_HTML.gif
        (3)
        So, we can obtain
        u j i = | a j i | + k j , i | a j k | m k i a j j | a j i | + k j , i | a j k | r i a j j = m j i , j i , j N , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equ4_HTML.gif
        (4)
        and
        u i m i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equan_HTML.gif
        Therefore, it is easy to obtain that
        a i i u i R i 1 + j i u j i a i i m i R i 1 + j i m j i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equao_HTML.gif
        Obviously, we have the desired result
        min i { a i i u i R i 1 + j i u j i } min i { a i i m i R i 1 + j i m j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equap_HTML.gif

         □

        Remark 3.1 According to inequalities (1) and (3), it is easy to know that
        b j i | a j i | + k j , i | a j k | m k i a j j b i i | a j i | + k j , i | a j k | d k a j j b i i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaq_HTML.gif
        and
        b j i | a j i | + k j , i | a j k | m k i a j j b i i | a j i | + k j , i | a j k | r i a j j b i i , i N . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equar_HTML.gif

        That is to say, the result of Lemma 2.2 is sharper than the ones of Theorem 2.1 in [8] and Lemma 2.2 in [9]. Moreover, the results of Theorem 3.2 and Theorem 3.3 are sharper than the ones of Theorem 3.1 in [8] and Theorem 3.3 in [9], respectively.

        Theorem 3.4 If A M n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq23_HTML.gif is strictly diagonally dominant by row, then
        τ ( A A 1 ) min i { 1 1 a i i j i | a j i | u j i } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equas_HTML.gif

        Proof This proof is similar to the one of Theorem 3.5 in [8]. □

        Remark 3.2 According to inequalities (2) and (4), we get
        1 1 a i i j i | a j i | u j i 1 1 a i i j i | a j i | s j i , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equat_HTML.gif
        and
        1 1 a i i j i | a j i | u j i 1 1 a i i j i | a j i | m j i . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equau_HTML.gif

        That is to say, the bound of Theorem 3.4 is sharper than the ones of Theorem 3.5 in [8] and Theorem 3.4 in [9], respectively.

        Remark 3.3 Using the above similar ideas, we can obtain similar inequalities of the strictly diagonally M-matrix by column.

        4 Example

        For convenience, we consider the M-matrix A is the same as the matrix of [8]. Define the M-matrix A as follows:
        A = [ 4 1 1 1 2 5 1 1 0 2 4 1 1 1 1 4 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equav_HTML.gif
        1. 1.
          Estimate the upper bounds for entries of A 1 = ( b i j ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq41_HTML.gif. Firstly, by Lemma 2.2(2) in [9], we have
          A 1 [ 1 0.5833 0.5000 0.5000 0.6667 1 0.5000 0.5000 0.5000 0.6667 1 0.5000 0.5833 0.5833 0.5000 1 ] [ b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaw_HTML.gif
           
        By Lemma 2.2, we have
        A 1 [ 1 0.5625 0.5000 0.5000 0.6167 1 0.5000 0.5000 0.4792 0.6458 1 0.5000 0.5417 0.5625 0.5000 1 ] [ b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 b 11 b 22 b 33 b 44 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equax_HTML.gif
        By Lemma 2.3 and Theorem 3.1 in [9], we get
        http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equay_HTML.gif
        By Lemma 2.3 and Lemma 3.1, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equaz_HTML.gif
        1. 2.

          Lower bounds for τ ( A A 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_IEq30_HTML.gif.

           
        By Theorem 3.2 in [9], we obtain
        0.9755 = τ ( A A 1 ) 0.8000 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equba_HTML.gif
        By Theorem 3.1, we obtain
        0.9755 = τ ( A A 1 ) 0.8250 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-65/MediaObjects/13660_2012_Article_499_Equbb_HTML.gif

        Declarations

        Acknowledgements

        This research is supported by National Natural Science Foundations of China (No. 11101069).

        Authors’ Affiliations

        (1)
        School of Mathematical Sciences, University of Electronic Science and Technology of China

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