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In this chapter we discuss how matrices interact with inner products and their induced norms and, occasionally, with more general norms. Our first task is to determine the most general inner product that can be placed on Euclidean space; this will lead us to the important notion of a symmetric positive definite matrix. We next discuss the Gram matrix construction, which enables one to readily construct positive definite matrices.

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How Matrices Interact with Inner Products and Norms

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  • 发表日期：

    2025年

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  • Jeff Calder,
  • Peter J. Olver

摘要

In this chapter we discuss how matrices interact with inner products and their induced norms and, occasionally, with more general norms. Our first task is to determine the most general inner product that can be placed on Euclidean space; this will lead us to the important notion of a symmetric positive definite matrix. We next discuss the Gram matrix construction, which enables one to readily construct positive definite matrices.