A Bauer-Hausdorff Matrix Inequality

Abstract

We present a biorthogonal process for two subspaces of ℂ^𝑛. Applying this process, we derive a matrix inequality, which generalizes the Bauer-Hausdorff inequality for vectors and includes the Wang-IP inequality for matrices. Meanwhile, we obtain its equivalent matrix inequality.

1. Introduction

The Cauchy-Bunyakovsky-Schwarz, or for short the C.B.S.-inequality, plays an important role in different branches of Modern Mathematics including Hilbert Space Theory, Probability & Statistics, Classical Real and Complex Analysis, Numerical Analysis, and Qualitative Theory of Differential Equations and their applications.

Given an 𝑛-dimensional complex space ℂ^𝑛 and two linear subspaces 𝑈 and 𝑉 such that 𝑈∩𝑉={0}, (1) there exists 𝛾=𝛾(𝑈,𝑉)∈[0,1) (2) such that for all 𝑥∈𝑈 and 𝑦∈𝑉 the following strengthened C.B.S.-inequality holds (see [1]): 𝑥^∗𝑦≤𝛾‖𝑥‖⋅𝑦, (3) where ‖⋅‖ denotes the standard Euclidian norm. The smallest quantity 𝛾 may be called the cosine of the angle 𝜙 between the spaces 𝑈 and 𝑉, or be called the C.B.S.-ratio of 𝑈 and 𝑉.

Strengthened C.B.S.-inequality has a long history and there exist various versions. The earliest result of this kind is due to the Wielandt [2] and was later generalized by many researchers. Among them two important extensions of the Wielandt inequality were given by Bauer and Householder [3], and Wang and Ip [4]. The aim of this paper is to present a matrix version of the Bauer-Hausdorff inequality like one of the Wielandt inequality given by Wang and Ip in [4].

On the practical side, this has been used in the analysis of two-level methods. The survey by Eijkhout and Vassilevski [1] attributes the basic theory of this inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations. Auzinger and Kirlinger [5] proposed another extension of this inequality for the resolvent conditions in the Kreiss matrix theorem [6].

Throughout the paper, we denote by 𝑅(𝑋) the range of the matrix 𝑋∈ℂ^𝑛×𝑝, 𝑅(𝑋)=𝑥∣𝑥=𝑋𝑧,𝑧∈ℂ^𝑝, (4) and by 𝑊(𝐴) the (closed) numerical range of an operator 𝐴 on the space ℂ^𝑛, 𝑊(𝐴)=𝑥^∗𝐴𝑥∣𝑥∈𝑅(𝐴),‖𝑥‖=1, (5) and by 𝐿(𝑎₁,𝑎₂,…,𝑎_𝑝) the linear subspace spanned by {𝑎₁,𝑎₂,…,𝑎_𝑝} with 𝑎_𝑖∈ℂ^𝑛, 𝑖=1,2,…,𝑝.

Definition 1. For 𝐴∈ℂ^𝑛×𝑛, 𝑃_𝐴=𝐴𝐴⁺ denotes the orthogonal projector onto the column space (range) of 𝐴, where 𝐴⁺ is the Moore-Penrose inverse of 𝐴.

Definition 2. For two 𝑛×𝑛 positive semidefinite Hermitian matrices 𝐴, and 𝐵, we say that 𝐴 is below 𝐵 with respect to the Löwner partial ordering, and we write 𝐴≤_𝐿𝐵, if 𝐵−𝐴 is positive semidefinite.

2. A Matrix Inequality

Let 𝐴 be an 𝑛×𝑛 positive definite Hermitian matrix. For any two nonzero complex vectors 𝑥∈𝑈 and 𝑦∈𝑉, Bauer and Householder [3] asserted that 𝑥^∗𝐴𝑦²𝑥^∗𝐴𝑥⋅𝑦^∗𝐴𝑦≤cos²𝜃, (6) where 𝜃 satisfies cot²𝜃2=𝜆₁𝜆_𝑛cot²𝜙2, (7) with 𝜙∈(0,𝜋/2] being the angle between two vector subspaces 𝑈 and 𝑉 mentioned above, and with 𝜆₁ and 𝜆_𝑛 being the largest and smallest, necessarily real and positive, eigenvalues of 𝐴.

A very interesting and important special case of (6) is the Wielandt inequality 𝑥^∗𝐴𝑦²𝑥^∗𝐴𝑥⋅𝑦^∗≤𝜆𝐴𝑦¹−𝜆_𝑛𝜆₁+𝜆_𝑛² (8) established by Wielandt [2] when 𝜙=𝜋/2. The upper bound in (8) is called the Wielandt ratio (see [7]). Wang and Ip [4] generalized this inequality as follows.

Let 𝑋 and 𝑌 be complex 𝑛×𝑝 and 𝑛×𝑞 matrices, respectively. If 𝑋^∗𝑌=0, then for all generalized inverses (𝑌^∗𝐴𝑌)⁻𝑋^∗𝐴𝑌𝑌^∗𝐴𝑌⁻𝑌^∗𝐴𝑋≤_𝐿𝜆₁−𝜆_𝑛𝜆₁+𝜆_𝑛²𝑋^∗𝐴𝑋 (9) in the Löwner partial ordering.

The first statistical application of the Wielandt inequality seems to be Eaton [8]. There are various equivalent versions of (8) in the literature. The most important one of them is the famous Kantorovich inequality [9, 10] which was used in estimating convergence rate of the steepest descent method for minimizing quadratic problems. Historical and biographical remark on them can be found in [7].

Lemma 3 (see Zhan, [11]). Let 𝐴∈ℂ^𝑛×𝑛 be a Hermitian matrix with 2×2 block form 𝐴𝐴=¹¹𝐴₁₂𝐴₂₁𝐴₂₂. (10) Then 𝐴 is positive semidefinite if and only if 𝐴₁₁ and 𝐴₂₂ are positive semidefinite and if there is a contraction matrix 𝑊 such that 𝐴₁₂=𝐴^1/2₁₁𝑊𝐴^1/2₂₂.

If 𝐴 can be partitioned as (10) and 𝐼𝑋=^𝑝00𝐼𝑌=_𝑛−𝑝, (11) then (9) becomes 𝐴₁₂𝐴⁻¹₁₁𝐴₂₁≤_𝐿𝜆₁−𝜆_𝑛𝜆₁+𝜆_𝑛²𝐴₂₂ (12) such that 𝑊^∗𝑊≤_𝐿𝜆₁−𝜆_𝑛𝜆₁+𝜆_𝑛²𝐼. (13) In other words, if 𝐴 is positive definite with the 2×2 block (10), then there is a contraction matrix 𝑊, with the maximum singular value being equal to or less than (𝜆₁−𝜆_𝑛)/(𝜆₁+𝜆_𝑛), such that 𝐴₁₂=𝐴^1/2₁₁𝑊𝐴^1/2₂₂.

The main result of this paper is stated in the following theorem.

Theorem 4. Let 𝐴∈ℂ^𝑛×𝑛 be a positive semidefinite Hermitian matrix with  rank𝑟≤𝑛 and eigenvalues 𝜆₁≥𝜆₂≥⋯≥𝜆_𝑟>0. For any 𝑋∈ℂ^𝑛×𝑝 and 𝑌∈ℂ^𝑛×𝑞, if 𝜙∈(0,𝜋/2] is the angle between the two vector spaces 𝑅(𝑃_𝐴𝑋) and 𝑅(𝑃_𝐴𝑌) and cot²𝜃2=𝜆₁𝜆_𝑟cot²𝜙2, (14) then 𝑋^∗𝐴𝑌𝑌^∗𝐴𝑌⁻𝑌^∗𝐴𝑋≤_𝐿cos²𝜃⋅𝑋^∗𝐴𝑋. (15)

Theorem 4 will be proved in the next section. This states a very general form of a (strengthened) C.B.S.-inequality and covers various types of C.B.S.-inequalities and their matrix forms. For instance, the inequality (15) reduces to (6) when 𝑝=𝑞=1 and reduces to (9) when 𝜙=𝜋/2.

Theorem 4 can be rewritten as the following equivalent form.

Theorem 5. Under the assumptions of Theorem 4, there is a contraction matrix 𝑊 (from Lemma 3), with the maximum singular value being equal to or less than cos𝜃, such that 𝑋^∗𝐴𝑌=𝑋^∗𝐴𝑋^1/2𝑊𝑌^∗𝐴𝑌^1/2. (16)

The maximum singular value of the contraction matrix 𝑊 in Theorem 5 might be strictly less than cos𝜃. For example, if 𝜆𝐴=cos𝜓−sin𝜓sin𝜓cos𝜓¹00𝜆₂=𝜆cos𝜓sin𝜓−sin𝜓cos𝜓₁cos²𝜓+𝜆₂sin²𝜓𝜆₁−𝜆₂𝜆sin𝜓cos𝜓₁−𝜆₂sin𝜓cos𝜓𝜆₁sin²𝜓+𝜆₂cos²𝜓, (17) then 𝑊=𝐴^−1/2₁₁𝐴₁₂𝐴^−1/2₂₂ such that 𝑊^∗𝜆𝑊=¹−𝜆₂²sin²𝜓cos²𝜓𝜆₁cos²𝜓+𝜆₂sin²𝜓𝜆₁sin²𝜓+𝜆₂cos²𝜓=𝜆₁−𝜆₂²sin²𝜓cos²𝜓𝜆₁−𝜆₂²sin²𝜓cos²𝜓+𝜆₁𝜆₂. (18) That is, the singular value of the contraction matrix 𝑊 belongs to the interval [0,(𝜆₁−𝜆₂)/(𝜆₁+𝜆₂)].

3. Proof of the Main Result

In this section we present an elementary proof of Theorem 4 by a biorthogonal procedure.

The C.B.S.-ratio 𝛾 in (3) can be redefined as 𝛾=max𝑥^∗𝑦∣‖𝑥‖=𝑦=1,𝑥∈𝑈,𝑦∈𝑉. (19) Since in finite dimensional spaces the unit sphere is compact, the maximum value of (19) is attained. The following result is obvious.

Lemma 6. Let 𝛾 be the C.B.S.-ratio of two subspaces 𝑈 and 𝑉 of ℂ^𝑛 satisfying (1). Then there exist two unit vectors 𝑢∈𝑈 and 𝑣∈𝑉 satisfying 𝛾=𝑢^∗𝑣 such that 𝑢−𝛾𝑣⟂𝑉,𝑣−𝛾𝑢⟂𝑈. (20)

One direct consequence of Lemma 6 is the following theorem.

Theorem 7. Let 𝑈 and 𝑉 be 𝑝 and 𝑞 dimensional linear subspaces of ℂ^𝑛 satisfying (1) with 𝑝,𝑞≥2 and 𝑝+𝑞≤𝑛. Then there exist two standard orthogonal bases {𝑢₁,𝑢₂,…,𝑢_𝑝} of 𝑈 and {𝑣₁,𝑣₂,…,𝑣_𝑞} of 𝑉 such that for each 𝑖=1,2,…,min{𝑝,𝑞}𝑢_𝑖⟂𝐿𝑣_𝑖+1,…,𝑣_𝑞,𝑣_𝑖⟂𝐿𝑢_𝑖+1,…,𝑢_𝑝. (21)

Proof. We shall achieve the desired result by the following biorthogonal process. Start with 𝑈₁=𝑈 and 𝑉₁=𝑉, 𝑖=1. By Lemma 6, one finds two unit vectors 𝑢_𝑖∈𝑈_𝑖 and 𝑣_𝑖∈𝑉_𝑖 such that 𝛾_𝑖=𝑢^∗_𝑖𝑣_𝑖,𝑢_𝑖−𝛾_𝑖𝑣_𝑖⟂𝑉_𝑖𝑣_𝑖−𝛾_𝑖𝑢_𝑖⟂𝑈_𝑖, (22)(23) where 𝛾_𝑖 is the C.B.S.-ratio of 𝑈_𝑖 and 𝑉_𝑖 with 𝛾_𝑖≤𝛾. If dim(𝑈_𝑖)=1 or dim(𝑉_𝑖)=1, then the procedure is completed; otherwise, update 𝑈_𝑖 and 𝑉_𝑖 by setting 𝑈_𝑖+1=𝑥𝑥−^∗𝑣_𝑖𝛾_𝑖𝑢_𝑖∣𝑥∈𝑈_𝑖,𝑉_𝑖+1=𝑦𝑦−^∗𝑢_𝑖𝛾_𝑖𝑣_𝑖∣𝑦∈𝑉_𝑖. (24) It is easily proved that 𝑢_𝑖⟂𝑉_𝑖+1,𝑣_𝑖⟂𝑈_𝑖+1. (25)
Replace 𝑖 by 𝑖+1, and repeat the above procedure until 𝑖=min{𝑝,𝑞}.
If 𝑝 is not equal to 𝑞, one finds a standard orthogonal bases {𝑢_𝑞+1,…,𝑢_𝑝} of 𝑈_𝑝 (𝑞<𝑝) or {𝑣_𝑝+1,…,𝑢_𝑞} of 𝑉_𝑝 (𝑝<𝑞). This procedure generates two bases {𝑢₁,𝑢₂,…,𝑢_𝑝} of 𝑈 and {𝑣₁,𝑣₂,…,𝑣_𝑞} of 𝑉 such that 𝑈_𝑖=𝐿(𝑢_𝑖,𝑢_𝑖+1,…,𝑢_𝑝) and 𝑉_𝑖=𝐿(𝑣_𝑖,𝑣_𝑖+1,…,𝑣_𝑞) for each 𝑖=1,2,…,min{𝑝,𝑞}. Equations (23) and (25) imply that 𝑢_𝑖⟂𝑈_𝑖+1,𝑣_𝑖⟂𝑉_𝑖+1, (26) such that these two bases are standard orthogonal. Finally, (21) holds since it is equivalent to (25).

In order to acquire our main result, we need the following lemmas.

Lemma 8. Let 𝑊(𝐴) be the numerical range of a linear operator 𝐴 on the space ℂ^𝑛. Then 𝑊𝑋^∗𝐴𝑋⊆𝑊(𝐴)⋅𝑊𝑋^∗𝑃_𝐴𝑋 (27) for any operator 𝑋∶ℂ^𝑘→ℂ^𝑛(𝑘∈ℕ arbitrary), where the multiplication is defined by 𝑋⋅𝑌={𝑥𝑦∶𝑥∈𝑋,𝑦∈𝑌}, 𝑋,𝑌∈ℂ.

Proof. For any 𝑧∈ℂ^𝑘, if 𝑃_𝐴𝑋𝑧=0, then 0=(𝑋^∗𝐴𝑋𝑧,𝑧)∈𝑊(𝐴)⋅𝑊(𝑋^∗𝑃_𝐴𝑋). If 𝑃_𝐴𝑋𝑧≠0 and ‖𝑧‖=1, then 𝐴=𝐴𝐴⁺𝐴𝐴⁺𝐴=𝑃_𝐴𝐴𝑃_𝐴 such that 𝑋^∗=𝑃𝐴𝑋𝑧,𝑧_𝐴𝑋𝑧²𝐴𝑃_𝐴𝑋𝑧𝑃_𝐴,𝑃𝑋𝑧^𝐴𝑋𝑧𝑃_𝐴(𝐴)𝑋𝑋𝑧∈𝑊⋅𝑊^∗𝑃_𝐴𝑋, (28) which results in the desired assertion.

When 𝐴 is invertible, Lemma 8 was established by Fujii [12].

Lemma 9. If 𝑝≤𝑞, then the matrix 𝐼𝐶=^𝑝𝐵𝐵^𝑇𝐼_𝑞 (29) has single eigenvalues 1±𝛾₁,…,1±𝛾_𝑝, where 𝐼_𝑝 and 𝐼_𝑞 are two 𝑝×𝑝 and 𝑞×𝑞 unit matrices, and 𝐵=(Λ,0)∈ℂ^𝑝×𝑞 with a 𝑝×𝑝 diagonal matrix Λ=diag(𝛾₁,𝛾₂,…,𝛾_𝑝). Furthermore, if 𝑝<𝑞, then 𝐶 has multiple eigenvalue 1.

Proof. If two vectors 𝜉=(𝜉₁,…,𝜉_𝑗,…,𝜉_𝑝+𝑞)^𝑇 and 𝜂=(𝜂₁,…,𝜂_𝑗,…,𝜂_𝑝+𝑞)^𝑇 are defined by 𝜉_𝑗=𝜂1,𝑗=𝑖,𝑝+𝑖,0,𝑗≠𝑖,𝑝+𝑖,_𝑗=1,𝑗=𝑖,−1,𝑗=𝑝+𝑖,0,𝑗≠𝑖,𝑝+𝑖 (30) then they are eigenvectors of 𝐶 with the eigenvalues 1+𝛾_𝑖 and 1−𝛾_𝑖, respectively.
If 𝑝<𝑞, for each 𝑗=2𝑝+1,…,𝑝+𝑞, the 𝑗th column vector of the (𝑝+𝑞)×(𝑝+𝑞) unit matrix is the eigenvector of 𝐶 with the multiple eigenvalue 1. The proof is completed.

Finally, we give the proof of Theorem 4.

Proof of Theorem 4. Let 𝑈 and 𝑉 be two standard orthogonal bases of ranges 𝑅(𝑃_𝐴𝑋) and 𝑅(𝑃_𝐴𝑌), respectively, with the ranks 𝑝 and 𝑞 in Theorem 7. Without loss of generality, we may assume that 𝑝≤𝑞. Let 𝑈^∗𝑉=(Λ,0)∈ℂ^𝑝×𝑞, where Λ= diag(𝛾₁,𝛾₂,…,𝛾_𝑝) is a 𝑝×𝑝 diagonal matrix with 𝛾_𝑖≤𝛾 for each 𝑖=1,2,…,𝑝 and 𝛾 the C.B.S.-ratio of 𝑅(𝑃_𝐴𝑋) and 𝑅(𝑃_𝐴𝑌). There exist two matrices 𝑋₁∈ℂ^𝑝×𝑝1 and 𝑌₁∈ℂ^𝑞×𝑞1 such that 𝑃_𝐴𝑋=𝑈𝑋₁ and 𝑃_𝐴𝑌=𝑉𝑌₁.
Letting 𝑍=(𝑈,𝑉), 𝐶=𝑍^∗𝑃_𝐴𝑍 can be expressed as the form (29) such that 𝑊(𝑍^∗𝑃_𝐴𝑍)=[1−𝛾,1+𝛾] from Lemma 9. Lemma 8 shows that the matrix 𝑄=𝑍^∗𝑈𝐴𝑍=^∗𝐴𝑈𝑈^∗𝑉𝐴𝑉_∗𝐴𝑈𝑉^∗𝐴𝑉 (31) has the largest and smallest eigenvalues 𝜆₁(1+𝛾) and 𝜆_𝑟(1−𝛾). Since (7) holds and 1+𝛾=1−𝛾1+cos𝜙1−cos𝜙=cot²𝜙2, (32) the Wielandt ratio of 𝑄 is 𝜆₁(1+𝛾)−𝜆_𝑟(1−𝛾)𝜆₁(1+𝛾)+𝜆_𝑟(1−𝛾)²=𝜆₁/𝜆_𝑟cot²𝜙/2−1𝜆₁/𝜆_𝑟cot²𝜙/2+1²=cot²𝜃/2−1cot²𝜃/2+1²=cos²𝜃 (33) such that the matrix 𝑈^∗𝐴𝑈𝑈^∗𝑉𝐴𝑉_∗𝐴𝑈cos²𝜃⋅𝑉^∗𝐴𝑉 (34) being positive semidefinite by the use of the Schur complement theory (see [13, 14]) to the inequality (9), which leads to the matrix 𝑋^∗𝐴𝑌𝑋^∗𝑌𝐴𝑌_∗𝐴𝑋cos²𝜃⋅𝑌^∗=𝑋𝐴𝑌^∗100𝑌^∗₁×𝑈^∗𝐴𝑈𝑈^∗𝑉𝐴𝑉_∗𝐴𝑈cos²𝜃⋅𝑉^∗×𝑋𝐴𝑉₁00𝑌₁ (35) is positive semidefinite. Applying the Schur complement theory again, the desired result is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported, in part, by the China Postdoctoral Science Foundation under Grant 2012M520417 and, in part, by the Natural Science Foundation of Fujian province of China under Grant 2012J01014. The authors are grateful to the anonymous referees for their careful comments and valuable suggestions on this paper.