A complement of the Hadamard-Fischer inequality

1 Introduction

Let M n be the set of n × n complex matrices. If X is positive semidefinite, we put X ≥ 0, for two Hermitian matrices X , Y ∈ M n , X ≥ Y means X - Y is positive semidefinite. If X is positive definite, we put X > 0. If X = [ X 11 X 12 X 21 X 22 ] with X₁₁ nonsingular, then the Schur complement of X₁₁ in X is defined by X / X 11 = X 22 - X 21 X 11 - 1 X 12 .

For A ∈ M n , the real and imaginary part of A are in the sense of the Cartesian decomposition [7, p. 7], they are denoted by R A = 1 2 ( A + A * ) , S A = 1 2 i ( A - A * ) , respectively.

Recall that the numerical range of A ∈ M n is defined by W ( A ) = { x * Ax | x ∈ ℂ n , x * x = 1 } .

Also, for θ ∈ [ 0 , π 2 ) , let S _θ be the sector in the complex plane given by S θ = { z ∈ ℂ | R z > 0 , | S z | ≤ ( R z ) tan ( θ ) } .

As 0 ∉ S _θ , if W (A) ⊂ S _θ , then A is necessarily nonsingular. Relevant studies on matrices with numerical ranges in a sector or complex matrices can be found in [2–4, 14].

Let A ∈ M n , for index sets α, β ⊂ {1, …, n}, we denote by A [α, β] the submatrix of entries that lie in the rows of A indexed by α and the columns indexed by β. If α = β, the submatrix A [α] = A [α, α] is a principal submatrix of A. If the index set α is empty, we adopt the convention that det A [α] =1. We denote by N (α) the cardinality of index set α.

The Hadamard inequality [7, p. 505] states that if A = [ a ij ] ∈ M n is positive definite, then det A ≤ a 11 ⋯ a nn .

Let A ∈ M n be partitioned as A = [ A 11 A 12 A 21 A 22 ] with A₁₁ square. If A is positive definite, then the Fischer inequality [7, p. 506] is det A ≤ det A 11 det A 22 .

The Hadamard and Fischer inequality above involve determinants of disjoint principal submatrices. The following Hadamard-Fischer inequality embraces the noncomplementary case: The principal submatrices are permitted to overlap.

Theorem 1.1. [7, p. 507] Let A ∈ M n be positive definite and let α, β ⊂ {1, …, n}. Then

( det A [ α ∪ β ] ) ( det A [ α ∩ β ] ) ≤ ( det A [ α ] ) ( det A [ β ] ) .(1)

For the following positive definite 3 × 3 block matrix H = [ H 11 H 12 H 13 H 21 H 22 H 23 H 31 H 32 H 33 ] ,(2) where the diagonal blocks are square and of arbitrary order, Ando and Petz [1] formulated a determinantal inequality as follows,

det H · det H 22 ≤ det [ H 11 H 12 H 12 * H 22 ] · det [ H 22 H 23 H 23 * H 33 ] .(3)

Recently, Lin and van den Driessche [12] gave the following inequality when the diagonal blocks of H are of the same order, which could be regard as a complement of (1.3). det H · det H 22 ≤ det [ H 11 H 12 H 12 * H 22 ] · det [ H 22 H 23 H 23 * H 33 ] - | det [ H 12 H 13 H 22 * H 23 ] | 2 .(4)

In this paper, by making use the idea of a proof of Lin and van den Driessche [12], we give an alternative proof and a following complement of the Hadamard-Fischer inequality.

Theorem 1.2. Let A ∈ M n be positive definite and let α, β ⊂ {1, …, n}. If N (α) = N (β), then

( det A [ α ∪ β ] ) ( det A [ α ∩ β ] ) ≤ ( det A [ α ] ) ( det A [ β ] ) - ∣ det A [ α , β ] ∣ 2 .(5)

Also, we present some results related to positive definite 3 × 3 block matrix and extend some relevant inequalities to a larger class of matrices, namely, matrices whose numerical range is contained in a sector.

In this section, we present some lemmas which are useful for our main results.

Lemma 2.1.[7, Theorem 7.7.7] Let F = [ A X X * B ] be Hermitian with A, B square. The following are equivalent:

F is positive definite.

Lemma 2.2. [5, Inequality (5.1)] Let [ A X X * B ] be positive semidefinite with all blocks square. Then det [ A X X * B ] ≤ det A · det B - ∣ det X ∣ 2 .

Lemma 2.3. [7, Theorem 7.8.19] Let A ∈ M n . If R ( A ) > 0 , then det ( R ( A ) ) ≤ | det ( A ) | .

Lemma 2.4. [9, Lemma 2.6] Let A ∈ M n with W (A) ⊂ S _θ . Then sec n ( θ ) det ( R ( A ) ) ≥ | det ( A ) | .

Lemma 2.5. [14, Property 2.6] Let A,  Q ∈ M _n , W (A) ⊂ S _θ , Q be an arbitrary nonsingular matrix. Then W (Q^*AQ) ⊂ S _θ . In particular, the numerical range of any matrix A ˆ , obtained by symmetric reordering of rows and columns in A, would be included in S _θ .

Lemma 2.6. [9, Proposition 2.1] Let A ∈ M n be partitioned as A = ( A 11 A 12 A 21 A 22 ) with A₁₁ square. If W (A) ⊂ S _θ , then W (A₁₁) ⊂ S _θ .

3 Main results

In this section, we begin with the alternative proof of Theorem 1.1 and the proof of Theorem 1.2.

Proof of Theorem 1.1. For s, t ⊂ {1, …, n}, in the proof, we denote A [s, s] and A [s, t] by A _s and A _st , respectively. Let s = α - β, t = β - α, r = α ∩ β, G = [ A s A st A sr A ts A t A tr A rs A rt A r ] .(6)

It can be derived that G is positive definite and det A [ α ∪ β ] = det G , det A [ α ∩ β ] = det A r .

If α∩ β = ∅, then det A [α ∩ β] =1, and G = [ A s A st A ts A t ] .(7)

Theorem 1.1 can be obtained by the Fischer inequality.

We consider α∩ β ≠ ∅. According to Lemma 2.1, the Schur complemen of A _r in G

G / A r = [ A s A st A ts A t ] - [ A sr A tr ] A r - 1 [ A rs A rt ] = [ A s - A sr A r - 1 A rs A st - A sr A r - 1 A rt A ts - A tr A r - 1 A rs A t - A tr A r - 1 A rt ](8) is positive definite. By the Fischer inequality, we have det ( G / A r ) ≤ det ( A s - A sr A r - 1 A rs ) det ( A t - A tr A r - 1 A rt ) .

Then det G det A r = ( det A r ) 2 det ( G / A r ) ≤ det A r det ( A s - A sr A r - 1 A rs ) · det A r det ( A t - A tr A r - 1 A rt ) = det [ A s A sr A rs A r ] det [ A t A tr A rt A r ] .

Note that det A [ α ] = det [ A s A sr A rs A r ] , det A [ β ] = det [ A t A tr A rt A r ] ,

Then ( det A [ α ∪ β ] ) ( det A [ α ∩ β ] ) = det G det A r ≤ ( det A [ α ] ) ( det A [ β ] ) .

This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2. We use the same notation as the proof of Theorem 1.1. If α∩ β = ∅, Theorem 1.2 can be derived by (3.2) and Lemma 2.2.

We consider α∩ β ≠ ∅. Note that G/A _r is positive definite and N (s) = N (t), by (3.3) and Lemma 2.2, we have det ( G / A r ) ≤ det ( A s - A sr A r - 1 A rs ) det ( A t - A tr A r - 1 A rt ) - ∣ det ( A st - A sr A r - 1 A rt ) ∣ 2 .

By det G = det A r det ( G / A r ) , we get det G det A r ≤ det A r det ( A s - A sr A r - 1 A rs ) · det A r det ( A t - A tr A r - 1 A rt ) - ( det A r ) 2 ∣ det ( A st - A sr A r - 1 A rt ) ∣ 2 = det [ A s A sr A rs A r ] det [ A t A tr A rt A r ] - ∣ det [ A st A sr A rt A r ] ∣ 2 .

Note that det A [ α , β ] = det [ A st A sr A rt A r ] .

Then det A [ α ∪ β ] det A [ α ∩ β ] = det G det A r ≤ ( det A [ α ] ) ( det A [ β ] ) - ∣ det A [ α , β ] ∣ 2 .

This completes the proof of Theorem 1.2. □

Remark 3.1. When N (α - β) = N (β - α) = N (α ∩ β), (1.5) could reduce to (1.4).

By (3.1) and the Fischer inequality, we get the following result.

Theorem 3.2. Let A ∈ M n be positive definite and let α, β ⊂ {1, …, n}. Then det A [ α ∪ β ] ≤ ( det A [ α - β ] ) ( det A [ α ∩ β ] ) ( det A [ β - α ] ) .

Now, we consider positive definite 3 × 3 block matrix with the diagonal blocks square. By Theorem 1.1, Theorem 1.2 and (3.1), we give the following results, which may be regarded as complements of the results of Ando, Petz, Lin and van den Driessche.

Theorem 3.3.Let H as defined in (1.2) be positive definite and let {u, v, w} = {1, 2, 3}. If the diagonal blocks of H are square, then det H · det H ww ≤ det [ H uu H uw H uw * H ww ] · det [ H vv H vw H vw * H ww ] .

Theorem 3.4.Let H as defined in (1.2) be positive definite matrix with the diagonal blocks square. Let {u, v, w} = {1, 2, 3}. If H _uu and H _vv are of the same order, then det H · det H ww ≤ det [ H uu H uw H uw * H ww ] · det [ H vv H vw H vw * H ww ] - | det [ H uv H uw H wv H ww ] | 2 .

Next, we show some generalizations of Theorems 1.1, 3.1 and 3.3 to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. Firstly, the generalization of Theorem 1.1 is shown as follows.

Theorem 3.5. Let A ∈ M n and let α, β ⊂ {1, …, n}. If W (A) ⊂ S _θ , then ∣ det A [ α ∪ β ] ∣ · ∣ det A [ α ∩ β ] ∣ ≤ sec N ( α ) + N ( β ) ( θ ) ∣ det A [ α ] ∣ · ∣ det A [ β ] ∣ .

Proof. According to Lemmas 2.5 and 2.6, we have A [α], A [β], A [α ∩ β], A [α ∪ β] ⊂ S _θ . Compute | det A [ α ∪ β ] | · | det A [ α ∩ β ] | ≤ sec N ( α ) + N ( β ) ( θ ) det ( R ( A [ α ∪ β ] ) ) · det ( R ( A [ α ∩ β ] ) ) ≤ sec N ( α ) + N ( β ) ( θ ) det ( R ( A [ α ] ) ) · det ( R ( A [ β ] ) ) ≤ sec N ( α ) + N ( β ) ( θ ) | det ( A [ α ] ) | · | det ( A [ β ] ) | , where the first inequality above is by Lemma 2.4; the second is due to Theorem 1.1 and the last inequality holds by Lemma 2.3. This completes the proof. □

When α = 0, Theorem 3.5 reduces to Theorem 1.1. A matrix A ∈ M n is accretive-dissipative if R A and S A are positive definie. For more details on this class of matrices, please refer to [6, 13]. Note that if A is accretive-dissipative, then W (e^-iπ/4A) ⊂ S_π/4. Thus, we get the following corollary.

Corollary 3.6. Let A ∈ M n and let α, β ⊂ {1, …, n}. If A is accretive-dissipative, then | det A [ α ∪ β ] | · | det A [ α ∩ β ] | ≤ 2 N ( α ) + N ( β ) 2 | det A [ α ] | · | det A [ β ] | .

Now, we give the generalizations of Theorems 3.1 and 3.3.

Theorem 3.7. Let A ∈ M n and let α, β ⊂ {1, …, n}. If W (A) ⊂ S _θ , then | det A [ α ∪ β ] | ≤ sec N ( α ∪ β ) ( θ ) | det A [ α - β ] | · | det A [ α ∩ β ] | · | det A [ β - α ] | .

Corollary 3.8. Let A ∈ M n and let α, β ⊂ {1, …, n}. If A is accretive-dissipative, then | det A [ α ∪ β ] | ≤ 2 N ( α ∪ β ) 2 | det A [ α - β ] | · | det A [ α ∩ β ] | · | det A [ β - α ] | .

Theorem 3.9.Let H ∈ M_n as defined in (1.2) be 3 × 3 block matrix with the diagonal blocks square. Let { u , v , w } = { 1 , 2 , 3 } , H ww ∈ M m . If W (H) ⊂ S _θ , then | det H | · | det H ww | ≤ sec m + n ( θ ) | det [ H uu H uw H uw * H ww ] | · | det [ H vv H vw H vw * H ww ] | .

Corollary 3.10.Let H ∈ M_n as defined in (1.2) be 3 × 3 block matrix with the diagonal blocks square. Let { u , v , w } = { 1 , 2 , 3 } , H ww ∈ M m . If H is accretive-dissipative, then | det H | · | det H ww | ≤ 2 m + n 2 | det [ H uu H uw H uw * H ww ] | · | det [ H vv H vw H vw * H ww ] | .