The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

中文翻译：

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任意界线性算子的奇异值展开

奇异值分解（SVD）是分析矩阵的基本工具。关于将矩阵定义为线性算子并为域和共域选择适当的正交基，可以将算符表示为对角矩阵。众所周知，SVD自然地扩展为将一个希尔伯特空间映射到另一个希尔伯特空间的紧凑线性算子。结果表示形式称为奇异值扩展（SVE）。众所周知，在希尔伯特空间上定义的一般有界线性算子也具有奇异值展开。该SVE允许对有关算子的各种问题进行简单分析，例如它是否定义了一个位置良好的线性算子方程式，以及在位置不正确时如何对方程进行正则化。