1 Introduction

Let \(\mathbb{M }_{n}(\mathbf C )\) be the set of \(n\times n\) complex matrices. For any \(A=(a_{ij})\in \mathbb{M }_n(\mathbf C )\), \(A^*\) stands for the conjugate transpose of \(A\). Similarly, \(x^*\) means the conjugate transpose of \(x\) for any \(x\in \mathbf C ^n\). \(A\in \mathbb{M }_{n}(\mathbf C )\) is accretive-dissipative if it can be written as \(A=B+iC\), where \(B=\frac{A+A^*}{2}\) and \(C=\frac{A-A^*}{2i}\) are both (Hermitian) positive definite. If \(B, C\) are real symmetric positive definite, then \(A\) is called a Higham matrix.Footnote 1

Consider the linear system

$$\begin{aligned} Ax=b \end{aligned}$$
(1)

and let \(A^{(k)} =(a^{(k)}_{ij})\) be the matrix resulted from applying the first \(k\) (\(1\le k\le n-1\)) steps of Gaussian elimination to \(A\). The quantity

$$\begin{aligned} \rho _n(A)\equiv \frac{\max _{i,j,k}|a_{ij}^{(k)}|}{\max _{i,j}|a_{ij}|} \end{aligned}$$

is called the growth factor (in Gaussian elimination) of \(A\). For more information on the numerical significance of investigating growth factor and its connection with the stability of Gaussian elimination, we refer to [3] and references threrein. It is proved in [3] that if \(A\) in (1) is a Higham matrix, then no pivoting is needed in Gaussian elimination. Higham [3] also conjectured that \(\rho _n(A)\le 2\) for such a matrix.

George et al. [2] made some progress concerning this conjecture. They obtained the following result:

Theorem 1.

Let \(A\in \mathbb{M }_{n}(\mathbf C )\) be accretive-dissipative. Then \(\rho _n(A)< 3\sqrt{2}\). If \(A\) is a Higham matrix, then \(\rho _n(A)< 3\).

They proved Theorem 1 via a stronger result, viz,

Theorem 2.

Let \(A\in \mathbb{M }_{n}(\mathbf C )\) be accretive-dissipative. Then

$$\begin{aligned} \frac{|a_{jj}^{(k)}|}{|a_{jj}|}<3, \quad j=1,\ldots , n; \quad k=1, \ldots , n-1. \end{aligned}$$
(2)

2 The Main Theorem

In this article, we show a tighter bound than (2). As a result, Theorem 1 is improved. Our result can be read as follows:

Theorem 3.

Let \(A\in \mathbb{M }_{n}(\mathbf C )\) be accretive-dissipative. Then

$$\begin{aligned} \frac{|a_{jj}^{(k)}|}{|a_{jj}|}<2\sqrt{2}, \quad j=1,\ldots , n; \quad k=1, \ldots , n-1. \end{aligned}$$
(3)

Consequently, \(\rho _n(A)< 4\). If \(A\) is a Higham matrix, then \(\rho _n(A)< 2\sqrt{2}\).

proof

Readers are assumed to have read [2]. The first few steps are the same as the proof in [2], so we skip them. We start from \(a_{jj}=b_{jj}+ic_{jj}\) and the fact that \(b_{jj}, c_{jj}>0\).

Setting

$$\begin{aligned} a_{jj}^{(k)}=\beta +i\gamma ,\quad \beta , \gamma \in \mathbf R , \end{aligned}$$

then we have

$$\begin{aligned} \beta =b_{jj}-b^*X_kb+c^*X_kc-2{\mathrm{Re }}(b^*Y_kc) \end{aligned}$$

and

$$\begin{aligned} \gamma =c_{jj}+b^*Y_kb-c^*Y_kc-2{\mathrm{Re }}(b^*X_kc), \end{aligned}$$

where

$$\begin{aligned} X_k=(B_k+C_kB_k^{-1}C_k)^{-1}\end{aligned}$$
(4)
$$\begin{aligned} Y_k=(C_k+B_kC_k^{-1}B_k)^{-1} \end{aligned}$$
(5)

with

$$\begin{aligned} \left[ \begin{array}{ll} B_k &{}\quad b \\ b^* &{}\quad b_{jj} \end{array}\right] {\quad } {\text{ and }} {\quad } \left[ \begin{array}{ll} C_k &{}\quad c \\ c^* &{}\quad c_{jj} \end{array}\right] \end{aligned}$$
(6)

positive definite. It is known that \(\beta ,\gamma >0\).

By the Cauchy-Schwarz inequality and the arithmetic mean-geometric mean inequality, we have

$$\begin{aligned} \pm \!2{\mathrm{Re }}(b^*Y_kc)\le 2\sqrt{(b^*Y_kb)(c^*Y_kc)}\le b^*Y_kb+c^*Y_kc; \end{aligned}$$
(7)
$$\begin{aligned} \pm 2{\mathrm{Re }}(b^*X_kc)\le 2\sqrt{(b^*X_kb)(c^*X_kc)}\le b^*X_kb+c^*X_kc. \end{aligned}$$
(8)

From (4) and (5) we have [2, Lemma 2.3]

$$\begin{aligned} X_k\le \frac{1}{2}C_k^{-1} {\quad } {\text{ and }} {\quad } Y_k\le \frac{1}{2}B_k^{-1}, \end{aligned}$$
(9)

where the inequality is in the sense of Loewner partial order. Also from (6), we know

$$\begin{aligned} b_{jj}>b^*B_k^{-1}b {\quad } {\text{ and }} {\quad } c_{jj}>c^*C_k^{-1}c \end{aligned}$$
(10)

Compute

$$\begin{aligned} |a_{jj}^{(k)}|&= |\beta +i\gamma |\\&\le \beta +\gamma \\&= b_{jj}-b^*X_kb+c^*X_kc-2{\mathrm{Re }}(b^*Y_kc)\\&+c_{jj}+b^*Y_kb-c^*Y_kc-2{\mathrm{Re }}(b^*X_kc)\\&\le b_{jj}-b^*X_kb+c^*X_kc+(b^*Y_kb+c^*Y_kc) {\quad }{({\text{ by } }} (7))\\&+c_{jj}+b^*Y_kb-c^*Y_kc+(b^*X_kb+c^*X_kc) {\quad } {({\text{ by } }} (8))\\&= b_{jj}+2b^*Y_kb+c_{jj}+2c^*X_kc\\&\le b_{jj}+b^*B_k^{-1}b+c_{jj}+c^*C_k^{-1}c {\quad }{({\text{ by } }} (9))\\&< 2(b_{jj}+c_{jj}) {\quad }{({\text{ by } }} (10))\\&\le 2\sqrt{2}|b_{jj}+ic_{jj}|\\&= 2\sqrt{2}|a_{jj}| \end{aligned}$$

This completes the proof of (3). To show the remaining claims, we need the following facts:

Fact 1. [2, Corollary 2.3] The property of being an accretive-dissipative matrix is hereditary under Gaussian elimination.

Fact 2. [2, Lemma 2.1, 2.2] If \(A=(a_{lj})\in \mathbb{M }_{n}(\mathbf C )\) is accretive-dissipative, then \(\sqrt{2}\max \limits _{l}|a_{ll}|\ge \max \limits _{l\ne j}|a_{lj}|\). If \(A\) is a Higham matrix, then \( \max \limits _{l}|a_{ll}|\ge \max \limits _{l,j}|a_{lj}|\).

Suppose \(\max _{j,k}|a_{jj}^{(k)}|=|a_{j_0j_0}^{(k_0)}|\) for some \(j_0, k_0\), then

$$\begin{aligned} \rho _n(A)=\frac{\max _{i,j,k}|a_{ij}^{(k)}|}{\max _{i,j}|a_{ij}|}\le \frac{\sqrt{2}\max _{j,k}|a_{jj}^{(k)}|}{\max _{i,j}|a_{ij}|}\le \frac{\sqrt{2}|a_{j_0j_0}^{(k_0)}|}{|a_{j_0j_0}|}<4. \end{aligned}$$

Similarly, we can show that if \(A\) is a Higham matrix, then \(\rho _n(A)<2\sqrt{2}\). The proof is thus complete. \(\square \)

We remark that Fact 2 in the preceding proof has been extended to norm inequalities for accretive-dissipative operator matrices; see [4].

3 Conclusion

Compared with Theorems 1 and 2, it might look minor to improve the upper bound from \(3\) to \(2\sqrt{2}\), but it is one step closer to the final solution of Higham’s conjecture. Moreover, the approach in the previous proof may also apply to other related results; see e.g. [1]. In [5], we have used a similar idea to improve a result on Fischer type determinantal inequalities for accretive-dissipative matrices.