By Eduard Jorswieck, Holger Boche
Majorization concept and Matrix-Monotone services in instant Communications, studies the elemental definitions of Majorization idea and Matrix-Monotone services, describing their innovations basically with many illustrative examples. as well as this educational, new effects are offered with recognize to Schur-convex features and concerning the houses of matrix-monotone features. The strategy taken via the authors presents a precious evaluation of the fundamental suggestions for readers who're new to the topic. They then continue to teach in separate chapters the innovative functions of the 2 uncomplicated theories in instant communications Majorization thought and Matrix-Monotone features in instant Communications is a useful source for college students, researchers and practitioners interested in the cutting-edge layout of instant communique platforms.
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Additional info for Majorization and Matrix Monotone Functions in Wireless Communications
15]. Let φ : [0, 1]2 → R be the function φ(x1 , x2 ) = log 1 −1 x1 + log 1 −1 . x2 Checking Schur’s condition it can be observed that φ is Schur-convex on x ∈ [0, 1]2 : x1 + x2 ≤ 1. However, the function log(1/t − 1) is convex on (0, 1/2] but not on [1/2, 1). 1 Inequalities in Matrix Theory There are also many applications of Majorization to matrix theory (see Chapter 9 in  and ). 9 (Schur inequality). Let H be an n × n Hermitian matrix. , λ(H) [H ii ]ni=1 . Proof. By the eigenvalue decomposition, we have H = U ΛU H .
1 Venn-diagram: Matrix-monotone functions are matrix-concave, concave, and monotone. 3 Fr´ echet Derivative Corresponding to the first and second derivatives of scalar functions, there exists a derivative of the matrix-valued function φ. We follow closely the derivation in [8, Sec. 3 and Sec. 4]. 4 (Fr´ echet differentiable). 3) holds. The linear operator Dφ(A)(H) is then called the derivative of φ at A in direction H. The difference from the scalar case is that a direction H is needed to define the derivative.
Hnn ] = [λ1 , . . , λn ]P = λP hence λ [H1 1, . . , Hn n]. 10 (Hadamard inequality). 5) k=l for all l = 1, . . , n with ordered eigenvalues λ1 (H) ≥ · · · ≥ λn (H). Equality holds if H is diagonal. Proof. 7 with g(x) = log x. g(x) = log(x) is a concave function and λ [H11 , . . 9. The eigenvalues of the sum of Hermitian matrices are characterized in many different ways . 2 Basic Results 21 proof that uses Majorization theory to give bounds on the spectrum of the sum of Hermitian matrices.
Majorization and Matrix Monotone Functions in Wireless Communications by Eduard Jorswieck, Holger Boche