Handbook of Linear Algebra, 2nd Edition

Chapter 5

Inner Product Spaces, Orthogonal Projection, Least Squares, and Singular Value Decomposition

Lixing Han

University of Michigan-Flint

Michael Neumann

University of Connecticut

5.1 Inner Product Spaces

Definitions:

Let V be a vector space over the field F, where F = ℝ or F = ℂ. An inner product on V is a function 〈·,·〉: V × V → F such that for all u, v, w ∈ V and a, b ∈ F, the following hold:

• 〈v, v〉 ≥ 0 and 〈v, v〉 = 0 if and only if v = 0.
• 〈au + bv, w〉 = a〈u, w〉 + b〈v, w〉.
• For F = ℝ: 〈u, v〉 = 〈v, u〉; For $F=ℂ:\text{〈}\mathbf{\text{u}},\mathbf{\text{v}}\text{〉}\text{=}\overline{\text{〈}\mathbf{\text{v}},\mathbf{\text{u}}\text{〉}}$ (where bar denotes complex conjugation).

A real (or complex) inner product space is a vector space V over ℝ (or ℂ), together with an inner product defined on it.

In an inner product space V, the norm, or length, of ...