Singular value inequalities and applications
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Positivity
Singular value inequalities and applications Wasim Audeh1 Received: 25 September 2020 / Accepted: 7 October 2020 © Springer Nature Switzerland AG 2020
Abstract It is shown among other inequalities that if A, B and X are n × n complex matrices such that A and B positive semidefinite, then s j (AX − X B) ≤ are 1 2 1/2 1 1 1/2 ∗ |X | A ⊕ 2 B + 21 B 1/2 |X |2 B 1/2 for j = 1, 2, . . . , 2n. sj 2A+ 2A Several related singular value inequalities and norm inequalities are also given. Keywords Concave function · Positive semidefinite matrix · Singular value · Unitarily invariant norm · Inequality Mathematics Subject Classification 15A18 · 15A42 · 15A60 · 47A30 · 47B15
1 Introduction Let Mn be the algebra of all n × n complex matrices. For A ∈ Mn , we denote the eigenvalues of |A| = (A∗ A)1/2 by s1 (A) ≥ s2 (A) ≥ · · · ≥ sn (A), they are called the singular values of A. Note that s j (A) = s j (A∗ ) = s j (|A|) for j = 1, 2, . . . , n. Note that the spectral (usual operator) norm . is the largest singular value, i.e. A = s1 (A), and the Schatten p-norms . p are defined interms of the singular 1/ p p n values, where A p = for 1 ≤ p ≤ ∞. Apart from the spectral j=1 s j (A) (usual operator) norm and the Schatten p-norms, we have the wider class of unitarily invariant norms |||.|||. Unitarily invariant norms are characterized by the invariance property which states that |||U AV ||| = |||A||| for all A ∈ Mn and for all unitary matrices U and V . Unitarily invariant norms are increasing functions of singular values (see, e.g., [4] or [9]). For A, B, X ∈ Mn , a matrix of the form AX − X A is called a commutator, a matrix of the form AX − X B is called a generalized commutator, a matrix of the form AX + X A is called anticommutator, and a matrix of the form AX + X B is called a
B 1
Wasim Audeh [email protected] Department of Mathematics, Petra University, Amman, Jordan
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W. Audeh
generalized anticommutator. In this paper, we present singular value inequalities for these types of matrices. Kittaneh in [11] has proved that if A, B ∈ Mn are positive semidefinite, then (1.1) s j (A + B) ≤ s j A + B 1/2 A1/2 ⊕ B + A1/2 B 1/2 for j = 1, 2, . . . , 2n. Inequality (1.1) can be extended to unitarily invariant norms. For j = 1, this inequality is the spectral norm inequality,
||A + B|| ≤ max A + B 1/2 A1/2 , B + A1/2 B 1/2 . (1.2) Specifying inequality (1.1) to the Schatten p-norms, we get p p 1/ p A + B p ≤ A + B 1/2 A1/2 + B + A1/2 B 1/2 p
p
(1.3)
for 1 ≤ p ≤ ∞. Kittaneh in [10] has been proved that if A, B ∈ Mn are positive semidefinite, then |||A + B||| ≤ |||A ⊕ B||| + A1/2 B 1/2 ⊕ A1/2 B 1/2 . (1.4) It should be mentioned here that inequality (1.4) is trivial consequence of inequality (1.1) by application of triangular inequality. Specifying inequality (1.4) to the spectral norm ., leads to ||A + B|| ≤ max { A , B} + A1/2 B 1/2 . (1.5) Davidson and Power in [8] has been shown a w
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