V2EX inner-product

Inner Product

定义 Definition

inner product(内积):线性代数与泛函分析中的一种运算,用来把两个向量(或函数)映射为一个标量,并满足正定性、对称性/共轭对称性、线性等性质。它常用于定义长度(范数)夹角(正交性)

发音 Pronunciation (IPA)

/nr prdkt/(美式常见)
/n prdkt/(英式常见)

例句 Examples

The inner product of two vectors can be zero if they are orthogonal.
如果两个向量正交,它们的内积可以为零。

In a complex vector space, the inner product uses conjugation so that x, x is always nonnegative, which lets us define a meaningful norm and notion of angle.
在复向量空间中,内积会用到共轭运算,从而保证 x, x 总是非负,这使我们能够定义有意义的范数与角度概念。

词源 Etymology

inner 意为“内部的/内在的”,product 在数学语境里常指“乘积/积”。“inner product”直译为“内(部)乘积”,用来强调它是在同一空间内部对两个元素进行的“乘法式”配对,并产生一个标量结果;这一术语在近代线性代数与希尔伯特空间理论发展中逐渐固定下来。

相关词 Related Words

文学与著作中的用例 Literary Works

  • Linear Algebra Done Right(Sheldon Axler)
  • Introduction to Linear Algebra(Gilbert Strang)
  • Linear Algebra(Kenneth Hoffman & Ray Kunze)
  • Principles of Mathematical Analysis(Walter Rudin,常在函数空间/内积相关内容中出现)
  • A Course in Functional Analysis(John B. Conway)
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