Linear Algebra and its Applications 327 (2001) 95–104 www.elsevier.com/locate/laa
On invertibility and positive invertibility of matrices M.I. Gil’ Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel Received 10 June 1999; accepted 20 September 2000 Submitted by R.A. Brualdi
Abstract New non-singularity and non-negative invertibility criteria for matrices are derived. They yield new bounds for the spectrum. Moreover, we obtain a bound for the norm of the inverse matrix and new estimates for the spectral radius. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A9; 15A18 Keywords: Matrices; Non-singularity; Non-negative invertibility; Bounds for spectra
1. Introduction Although excellent computer software are now available for eigenvalue computation, new results on invertibility and spectrum inclusion regions for finite matrices are still important, since computers are not very useful, in particular, for analysis of matrices dependent on parameters. But such matrices play an essential role in various applications, for example in stability and boundedness of coupled systems of partial differential equations, cf. [9, Section 14]. In addition, bounds for eigenvalues of finite matrices allow us to derive bounds for spectra of infinite matrices. Because of this, the problem of finding of invertibility conditions and spectrum inclusion regions for finite matrices continues to attract attention of many specialists, cf. [1,2,4,5,10,13,15] and references given therein. E-mail address:
[email protected] (M.I. Gil’). 0024-3795/01/$ - see front matter 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 0 0 ) 0 0 3 1 3 - X
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M.I. Gil’ / Linear Algebra and its Applications 327 (2001) 95–104
Let A = (aj k ) be a complex n × n matrix (n 2) with the non-zero diagonal: akk = / 0 (k = 1, . . . , n). Put n Pj = |aj k |. k=1, k =j /
The well-known Levy–Desplanques theorem states that if |ajj | > Pj (j = 1, . . . , n), then A is nonsingular. This theorem has been improved in many ways. For example, each of the following is known to be a sufficient condition for non-singularity of A: (i) |aii ||ajj | > Pj Pi (i, j = 1, . . . , n) [14, p. 149]. (ii) |ajj | Pj (j = 1, . . . , n), provided that at least one inequality is strict and A is irreducible [14, p. 147]. (iii) |ajj | rj mj (j = 1, . . . , n), where rj are positive numbers satisfying n (1 + rk )−1 1 and mj = max |aj k | k =j /
k=1
(see [1] and references therein).
(iv) |ajj | > Pj Q1− (j = 1, . . . , n), where j n
Qj =
0 1,
|akj |
[14, p. 150].
k=1, k =j /
(v) |ajj | > Rj (j = 1, . . . , n), where Rj =
j −1 n Rk |aj k | + |aj k | (j > 1), |akk |
R1 = P1
[10].
k=j +1
k=1
Note that for any of above-mentioned well-known criteria one can give such a triangular matrix with a non-zero diagonal that the corresponding criterion is not applied. A sufficient condition for invertibility, which is always satisfied by an arbitrary upper triangular matrix with a non-zero diagonal, was derived in [7]. Namely, let n
Sm =
|aj m |,
m = 1, . . . , n − 1,
Sn = 0,
j =m+1
and µ(A) ≡
1 +
3p n
1j 0 for all j = 1, . . . , n
(1.4)
k=1
hold. Then by the Collatz theorem [3, p. 377], A is monotone. In other words, A is non-negatively invertible or M-matrix, cf. [11]. In [8], it was shown that A is Mmatrix if conditions (1.1) and (1.2) hold. In the present paper, the new non-negative invertibility conditions are obtained. They supplement conditions (1.3) and (1.4).
2. Statement of the main result Take the norm n
A ∞ ≡ max
j =1,...,n
|aj k |.
k=1
Throughout the present and the following sections it is assumed that d0 ≡ min |ajj | > 0. k=1,...,n
Introduce the notation v˜k = w˜ k =
max
|aj k | (k = 2, . . . , n);
max
|aj k | (k = 1, . . . , n − 1),
j =1,...,k−1
j =k+1,...,n
mup (A) ≡
n v˜k 1+ |akk |
k=2
and mlow (A) =
n−1
1+
k=1
w˜ k . |akk |
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M.I. Gil’ / Linear Algebra and its Applications 327 (2001) 95–104
Theorem 2.1. Let the condition mup (A)mlow (A) < mup (A) + mlow (A)
(2.1)
be fulfilled. Then matrix A is invertible and the inverse matrix satisfies the inequality A−1 ∞
mup (A)mlow (A) . (mup (A) + mlow (A) − mup (A)mlow (A))d0
(2.2)
Moreover, under conditions (1.2) and (2.1) matrix A−1 is non-negative. The proof of this theorem is divided into a series of lemmas which are presented in the following section. Note that condition (2.1) is equivalent to the following one: θ (A) ≡ (mup (A) − 1)(mlow (A) − 1) < 1.
(2.3)
Inequality (2.2) can be written as A−1 ∞ ≤
mup (A)mlow (A) . d0 (1 − θ (A))
(2.4)
Let matrix A have a non-zero diagonal and triangular triangular. Then (2.1) obviously holds. Theorem 2.1 supplements the Levy–Desplanques theorem, the Collatz one and other existing results. Note that in the case n = 2, according to Theorem 2.1, the condition |a21 a12 | < |a22 a11| provides the invertibility, while the Levy–Desplanques theorem requires the inequalities |a21 | < |a22| and |a12 | < |a11 |.
3. Proof of Theorem 2.1 In this section, it is assumed that all the diagonal entries are non-zero and · is an arbitrary matrix norm with the properties I = 1, where I is the identity matrix, and BC B C for arbitrary matrices B, C. Denote by V , W and D the upper triangular, lower triangular, and diagonal parts of matrix A = (aj k )nj,k=1 , respectively. That is, 0 a12 . . . a1n 0 ... 0 0 0 0 . . . a2n 0 0 , W = a21 . . . V = . . . . . . . ... . 0 0 ... 0 an1 . . . an,n−1 0 and D = diag[a11, a22 , . . . , ann ]. So, A = D + V + W . Set WA = W D −1
and VA = V D −1 .
Lemma 3.1. Let the condition
n−1
j k+j k ˜θ0 ≡ (−1) VA WA
