Further extensions of Hartfiel’s determinant inequality to multiple matrices


 Following the recent work of Zheng et al., in this paper, we first present a new extension Hartfiel’s determinant inequality to multiple positive definite matrices, and then we extend the result to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. Our result complements that of Mao.


Introduction
Throughout the paper, we denote by Mn the set of n × n complex matrices. Recall that the numerical range (see, e.g., [3]) of A ∈ Mn is de ned as the set on the complex plane For a xed θ ∈ [ , π/ ), obviously the set on the complex plane S θ = {z ∈ C : z > , | z| ≤ ( z) tan θ} is a sector excluding the vertex. We shall mainly consider matrices whose numerical range is contained in S θ , the so called sector matrices [11]. For any A ∈ Mn, its real (or Hermitian) part is denoted by A := (A + A * )/ . Clearly, if W(A) ⊂ S θ , then A is positive de nite.
A fundamental determinant inequality states that if A, B ∈ Mn are positive de nite, then In [5], Haynsworth proved the following improvement of (1).
Using a clever argument, Hart el [4] re ned Haynsworth's result by adding a nonnegative term on the right side of the inequality.
Hart el's inequality (2) has been extended to sector matrices by a number of authors; see [8,10,14,17]. In [8], Hou and Dong extended Hart el's inequality to a triple of matrices. By making use of Hou and Dong's result, Zheng et al. [17] improved and extended the main result in [10], moreover, they obtained the following two theorems.
where by convention, det M = det N = det L = .
Very recently, Mao in [14] extended Theorem 1.3 to any number of sector matrices. More precisely, she obtained the following result.
The main goal of the present paper is to extend Theorem 1.5 to any number of sector matrices. To this end, we rst present a relevant result for positive de nite matrices. Some corollaries are included.
The rst lemma is folklore in matrix analysis.
where by convention det A = det B = .

Lemma 2.2. Let A ∈ Mn with A positive de nite. Then
The third lemma gives a reverse of the Ostrowski-Taussky inequality.

Lemma 2.3. [10, Lemma 2.6] Let A ∈ Mn with W(A) ⊂ S θ . Then
The fourth lemma pays a key role in our new extension of Hart el's inequality to multiple positive de nite matrices.
Taking products for k from to n gives Now by Lemma 2.4, we have This completes the proof.
The following result extends Theorem 1.5.
Theorem 2.6. Let A j ∈ Mn, j = , . . . , m, such that W(A j ) ⊂ S θ , and let A jk , k = , . . . , n − , denote the kth leading principal submatrix of A j . Then it holds where by convention det A j = for all j.
Proof. First of all, since W( m j= A j ) ⊂ S θ , then by Lemma 2.2, As A j are positive de nite for all j, we can apply Proposition 2.5 to get in which the second inequality is by Lemma 2.2 and the third inequality is by Lemma 2.3, respectively. This completes the proof.
A matrix A ∈ Mn is called accretive-dissipative if both real part A and imaginary part A := (A − A * )/ i (in the sense of Cartesian decomposition) are positive de nite. This class of matrices has appeared in numerical linear algebra [2,6,12] and has been studied recently by a number of authors [9,15,16] where by convention det A j = for all j.