Matrix Analysis

By Rajendra Bhatia. 1996

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Citations in the corpus

A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks (2018, neutral): “Lemma 3 (Bhatia (1997), Exercise IV.3.5) .”
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions (2009, neutral): “Standard references for this material include [11, 72].”
Asymptotic Error Rates in Quantum Hypothesis Testing (2007, methodology): “lowing inequality holds: Tr[PB(At−Bt)] ≥ 0. (25) Asymptotic Error Rates in Quantum Hypothesis Testing 15 Proof. To deal with the t-th matrix power, we use an integral representation (see, for example [3] (V.56)). For scalars a≥ 0and 0 ≤ t≤ 1, at= sin(tπ) π Z+∞ 0 dxxt−1 a a+x . For other values of tt”
Analyze gauss: optimal bounds for privacy-preserving principal component analysis (2014, background): “In the following useful result from matrix perturbation theory (see, for example [3]) we will denote λ 1 ( X ) ≥ λ 2 ( X ) · · · as the eigenvalues of a symmetric matrix X .”
Operator Valued Hardy Spaces (2003, unknown): “ts e ε ∈ P(M)such that τ(1 −e ε) <εand lim λ→λ 0 ke ε(f λ −f)e εk∞ = 0. Obviously, f λ a.u→ fimplies f λ b.a.u→ f. Recall that the map x→ xp (1 ≤ p≤ 2) is convex on the positive cone M + of M (see [2]). Thus, for f∈ Lp(L∞(R) ⊗M) (1 ≤ p≤ 2),we get (3.15) Z A |f|dt≤ ( Z A |f|pdt)1p , ∀A⊆ R, |A| = 1”
Cauchy–Schwarz norm inequalities for weak∗-integrals of operator valued functions (2005, background): “It is the arithmetic–geometric–logarithmic (56),(74) and Young (66) inequalities that have been under especially intensive investigation and a very good account can be found in [2,6,16].”