Some new two-sided bounds for determinants of diagonally dominant matrices

In this article, we present some new two-sided bounds for the determinant of some diagonally dominant matrices. In particular, the idea of the preconditioning technique is applied to obtain the new bounds.

MS Classification: 65F10; 15A15.

By we denote the set of all n × n complex (real) matrices. A matrix is called a Z-matrix if a_ij ≤ 0 for any i ≠ j; a nonsingular M-matrix if A is a Z-matrix with A^-1 is nonnegative, i.e., A^-1 ≥ 0. The comparison matrix 〈A〉 = (ã_ij) for A is defined by

where 〈n〉 ≡ {1, 2,..., n}.

Throughout this article, we always assume that A = D - L - U, where D, -L and -U are nonsingular diagonal, strict lower and strict upper triangular parts of A. It is noted that 〈A〉 = |D| - |L| - |U|, where |C| = (|c_ij|) for C = (c_ij).

Let and Λ_i(B) = Σ_{i≠k∈〈n〉} |b_ik|. Then it is easy to see that 〈A〉e = (|a₁₁| - Λ₁(A),..., |a_nn| - Λ_n(A))^T, where e = (1,..., 1)^T with appropriate dimensions. Let , and . Then |L|e = (l₁,...,l_n), |U|e = (u₁,...,u_n), and Λ_i(A) = l_i + u_i.

Definition 1.1 Let . Then A is said to be

(1) a diagonally dominant matrix (d.d.), if |a_ii| ≥ Λ_i(A) for each i ∈ 〈n〉;

(2) a strictly diagonally dominant matrix (s.d.d.), if |a_ii| > Λ_i(A) for each i ∈ 〈n〉;

(3) a weakly chained diagonally dominant matrix (c.d.d.) (e.g., see [1,2]), if A is a d.d. matrix with , where β(A) = {j | |a_jj| > Σ_{j≠k∈〈n〉} |a_jk|) and for all i ∈ 〈n〉, i ∉ β(A), there exist indices i₁,...,i_k in 〈n〉 with , 0 ≤ r ≤ k -1, where i₀ = i and i_k ∈ β(A).

(4) a generalized diagonally dominant matrix (g.d.d.), if there is a positive diagonal matrix D such that AD is an s.d.d. matrix.

It is noted that the comparison matrix of a g.d.d. matrix is a nonsingular M-matrix (e.g., see [[1], Lemma 3.2]).

The classical bound for the determinant of an s.d.d. matrix A is the Ostrowski's inequality [3], i.e.,

which was improved by Price as follows [4]

The bound (1.1) was further improved by Ostrowski [5] and Yong [6]. In [6] the author obtained the following two-sided bounds for s.d.d. matrices (see [[6], Theorem 2.2])

The inequalities of the determinant can be applied to estimate the spectral of a matrix and to determine the nonsingularity of a matrix, etc, which are useful in numerical analysis. Some numerical examples show that the bound in (1.2) is not optimal. By this motivation, in this article, we consider to give some sharper bounds than the ones in (1.1) and (1.2). The rest of this article is organized as follows. In Section 2, we use the classical technique to obtain new two-sided bounds; see Theorems 2.5 and 2.5'. In Section 3, we apply the idea of the preconditioning technique to give a new bound for the M-matrix case; see Theorem 3.2. A conclusion is given in the final section.

Let α₁ and α₂ be two subsets of 〈n〉 such that 〈n〉 = α₁⋃α₂ and . Let . By A_ij = A[α_i|α_j] we denote the submatrix of A whose rows are indexed by α_i and columns by α_j. For simplicity, we denote A[α₁] instead of A[α_i|α_i]. If A[α₁] is nonsingular, the Schur complement of A[α₁] in A is denoted by , i.e., . By A_(k) we denote A_(k) = A[α^(k)], where α^(k) = {k + 1,..., n}.

We define s_k(A) as follows:

Alternatively, the recursive Equation (2.1) can be computed by the following lemma, which can be deduced from the similar proof to those in [7].

Lemma 2.1 Let . Then

The following lemma is well-known, e.g., see [1].

Lemma 2.2 Let A be a c.d.d. matrix. Then A is g.d.d., and hence is nonsingular.

Now we partition A into the following block form:

Then it is easy to check that

where .

The following lemma can be found in [8].

Lemma 2.3 Let A = (a_ij) be a nonsingular d.d. M-matrix, and let . Then

Lemma 2.4 Let A be a c.d.d. matrix. Then

where we define A₍₀₎ = A.

Proof. It follows from Lemma 2.2 that A is nonsingular. Let A be as in (2.3) and . Then we have

By (2.4) we have

which together with (2.7) gives that

It follows from (2.5) and (2.8) that

Because A is c.d.d., 〈A〉 is a nonsingular M-matrix, and so is 〈A₍₁₎〉, which implies that A₍₁₎ is also a c.d.d. matrix (see [ 1, Theorem 3.3]). Applying the induction on k to (2.9) one may deduce the desired inequality (2.6).

Remark 2.1 It is difficult to compute the bound (2.6) because one needs to compute all s_i(A_(k-1)), i = n,...,k for k = 1,...,n. However, we may replace s₁(A_(k-1)) by s_i(A).

Theorem 2.5 Let A be a c.d.d. matrix. Then

Proof. By (2.1) we have

and hence

Therefore, we have

which together with (2.6) gives the bound (2.10).

Let . Then R_k(A) is given by (e.g., see [9] or [1])

A matrix A is called a Nekrasov matrix ([9] or [1]) if |a_kk| > R_k(A) for k ∈ 〈n〉. A Nekrasov matrix is a g.d.d. matrix (e.g., see [9]). The bound for the determinant of a Nekrasov matrix is given below (see [10,11]):

However there is a typos for this bound, a counter-example was given in [12]. In the following theorem, we get an estimation of the determinant of A by using R_i (A), the proof is analogical to those in Theorem 2.5.

Theorem 2.5' Let A be a c.d.d. matrix. Then

Remark 2.2 Let A = D - L - U. Then the recursive Equations (2.1) and (2.11) for S_k(A) and R_k(A) can be computed by (2.2) and the following formula (see [7])

respectively. Hence two bounds (2.10) and (2.13) are based on different splittings A = (D - U) - L = (D - L) - U. The following two examples illustrate that none of these two bounds is better than other.

Example 2.1 Let

Then A is an s.d.d. matrix. Applying the bounds (2.10) and (2.13) to this matrix yields

and

respectively, which shows that the bound in (2.13) is better.

Example 2.2 Let

Then A is s.d.d.. By (2.10) and (2.13), we have

and

respectively. Hence the bound (2.10) is better.

Remark 2.3 It is noted that the bound in (2.10) (or (2.13)) only provides alternative estimation for the determinant, this bound does not improve (1.2) in general. However, Example 2.1 illustrates that the bound in (2.10) is better. In fact, by (1.2) we obtain

The following example shows that the upper bound in (1.2) can be better than the one in (2.10).

Example 2.3 Let

Then by (2.10) and (1.2) we have

and

respectively.

It is well known that the preconditioning technique plays more and more important roles in solving linear systems (e.g., see [13]). In this section we improve the bound (1.2) based on the idea of preconditioning.

Without loss of generality we may assume that all diagonal entries of A are equal to 1 in this section. Otherwise, we consider the matrix D^-1A, where D = diag(a₁₁,..., a_nn). Then det(D^-1A) = det D^-1 det A Hence, we assume that

where L and U are a strictly lower triangular and a strictly upper triangular matrices, respectively

Let

which was first introduced in [14] for solving linear systems, and was further studied by many authors (e.g., see [15-18]). Usually, P is call a preconditioner for solving the linear system Ax = b.

Let B = PA. Then det B = det A and

where and are a lower triangular and a strictly upper triangular matrices, respectively. The ith diagonal entry of B is given by

If A is an s.d.d. M-matrix, so is B (see [16]). Let A have the block form (2.3). We partition I + S into the following block form

where and . A simple calculation yields that

where

Then

It is easy to see that

By (3.3) we have

and hence from [19] (also see [6]) it follows that

Notice that , which together with (3.4), (3.5), (3.6), and (3.2) gives

By (3.3), B₍₁₎ is also the preconditioned matrix of A₍₁₎ with the preconditioner I + S₍₁₎. In this case, B₍₁₎ is also an s.d.d. matrix. So we may proceed by induction with (3.7), and then one may easily deduce the following lemma.

Lemma 3.1 Let A be an s.d.d. M-matrix with unit diagonal entries. Then

By the above argument, we may deduce the following result without the assumption that A has unit diagonal entries as in Lemma 3.1.

Theorem 3.2 Let A be an s.d.d. M-matrix. Then

Remark 3.1 It is noted that the bound (3.9) is always sharper than the one (1.2). In fact, for any i, u_i < |a_ii| we have and hence the upper bound is better than the one in (1.2). For the lower bound, since

the lower bound in (3.9) is better than the one in (1.2), which proves our assertion.

Remark 3.2 None of these two bounds in (3.9) and (2.10) is uniformly better than other. However the following example illustrates that the upper bound in (3.9) is better.

Example 3.1 Let

Applying (3.9), (1.2), and (2.10) to estimate the determinant of A, respectively we have

and

In Sections 2 and 3, we have provided some two-sided bounds for the determinant of a d.d. matrix via both classical and preconditioning techniques. Although none of two bounds in (1.2) and (2.10) are uniformly better than other in general, the condition in the (2.10) is weaker than the one in (1.2).

When the preconditioning technique is applied to estimate the determinant of an s.d.d. M-matrix, we may obtain a more tighter bound. Here, we only present a bound (3.9) for the special preconditioner (3.1), and prove that this bound is sharper than the bound (1.2), which shows that a good preconditioning technique is a powerful tool not only for solving linear system but also for some estimations such as determinants etc.

The authors declare that they have no competing interests.

Both authors participated the research study of this article, and read and approved the finial manuscript.

The work was supported in part by National Natural Science Foundation of China (No. 10971075), Research Fund for the Doctoral Program of Higher Education of China (No. 20104407110002) and Guangdong Provincial Natural Science Foundation (No. 9151063101000021), P.R. China.

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