Titlebook: Loewner‘s Theorem on Monotone Matrix Functions; Barry Simon Book 2019 Springer Nature Switzerland AG 2019 matrix convex.approximation theo

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不能妥协

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重力

Introduction: The Statement of Loewner’s TheoremMost of this monograph will consider finite matrices, but since we will be interested in positivity defined in terms of a (Euclidean) inner product, we make definitions in a general context.

lipoatrophy

Herd-Immunity

The Herglotz Representation Theorems and the Easy Direction of Loewner’s TheoremIn this chapter, we will show (b)⇔(c) in Theorems . and . and also (b)⇒(a) and the equivalent part of Theorem .. We’ll have a different proof of (c)⇒(a) in Chapter .. We want to study functions . on . with ., but begin with functions . on . with . and use conformal mapping.

Metastasis

Monotonicity of the Square RootAs explained in the preface, most applications of Loewner’s theorem involve the easy half of the theorem. This chapter is an aside involving the two most significant and common matrix monotone functions: fractional powers and the log.

公猪

Loewner MatricesIn this chapter, we will reduce matrix monotonicity to the positivity of certain matrices and determinants. Seven of the eleven proofs we’ll give of the hard part of Loewner’s theorem rely on this reduction.

declamation

gregarious

In this chapter, we’ll present explicit exampIn this chapter, we’ll present explicit examples that show that . is strictly smaller than . and that show that 2. − 3 in Theorem . is optimal.

单色

Heinävaara’s Second Proof of the Dobsch–Donoghue TheoremIn this chapter, we provide a really short, simple, and elegant proof due to Heinävaara of Theorem .. It will be a direct consequence of the mean value theorem for divided differences. Below, . will denote the real polynomials of degree at most ., a real vector space of dimension . + 1.