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Mathematics for Physics

255 Inner Product

A mapping that assigns a scalar to each ordered pair of vectors that is used to define length through a norm and to compare vectors.

definition [d] (Inner Product = Scalar Product = Dot Product) A map \((\cdot,\, \cdot) : V \times V \rightarrow K\) associating a scalar with every ordered pair of vectors, satisfying for all \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in V\) and scalars \(\alpha, \beta\):

  • (Conjugate symmetry) \((\mathbf{u},\, \mathbf{v}) = \overline{(\mathbf{v},\, \mathbf{u})}\) .
  • (Linearity in the first argument) \((\alpha\mathbf{u} + \beta\mathbf{v},\, \mathbf{w}) = \alpha(\mathbf{u},\, \mathbf{w}) + \beta(\mathbf{v},\, \mathbf{w})\) .
  • (Positive-definiteness) \((\mathbf{u},\, \mathbf{u}) \geq 0\), and \((\mathbf{u},\, \mathbf{u}) = 0 \iff \mathbf{u} = 0\) .

where

  • \(V\) is a vector space over a field \(K\).
  • \(\overline{\,\cdot\,}\) denotes the complex conjugate.

Note:

  • \(K\) is typically \(\mathbb{R}\).
  • \(K\) is also typically \(\mathbb{C}\).
  • the complex conjugate is redundant if \(K = \mathbb{R}\).

definition [d] (Inner Product = Scalar Product = Dot Product) A map \(\langle \cdot,\, \cdot \rangle : V \times V \rightarrow K\) associating a scalar with every ordered pair of vectors, satisfying for all \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in V\) and scalars \(\alpha, \beta\):

  • (Conjugate symmetry) \(\langle \mathbf{u},\, \mathbf{v} \rangle = \overline{\langle \mathbf{v},\, \mathbf{u} \rangle}\) .
  • (Linearity in the second argument) \(\langle \mathbf{u},\, \alpha\mathbf{v} + \beta\mathbf{w} \rangle = \alpha\langle \mathbf{u},\, \mathbf{v} \rangle + \beta\langle \mathbf{u},\, \mathbf{w} \rangle\) .
  • (Positive-definiteness) \(\langle \mathbf{u},\, \mathbf{u} \rangle \geq 0\), and \(\langle \mathbf{u},\, \mathbf{u} \rangle = 0 \iff \mathbf{u} = 0\) .

where

  • \(V\) is a vector space over a field \(K\).
  • \(\overline{\,\cdot\,}\) denotes the complex conjugate.

Note:

  • \(K\) is typically \(\mathbb{R}\).
  • \(K\) is also typically \(\mathbb{C}\).
  • the complex conjugate is redundant if \(K = \mathbb{R}\).

255.1 Elementary Example

255.1.1 Simple

An inner product assigns a scalar to each ordered pair of vectors. On \(\mathbb{R}^{2}\) use the dot product.

\[ \langle u,v \rangle = u_{1} v_{1} + u_{2} v_{2} \]

\[ u = (1,2),\quad v = (3,4),\quad \langle u,v \rangle = 11 \]

where

  • \(\langle \cdot,\cdot \rangle\) is the inner product.
  • \(u, v\) are vectors.

255.1.2 General

On \(\mathbb{C}^{3}\), the standard Hermitian inner product conjugates the second factor.

\[ \langle u,v \rangle = u_{1}\overline{v_{1}} + u_{2}\overline{v_{2}} + u_{3}\overline{v_{3}} \]

\[ \langle u,u \rangle \ge 0,\quad \langle u,u \rangle = 0 \iff u = 0 \]

where

  • \(\overline{v_{j}}\) is the complex conjugate of the component \(v_{j}\).

255.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — inner-product axioms (linearity in the first argument).
  2. Griffel, D. H. Applied Functional Analysis. Ellis Horwood, 1981. — inner-product axioms.
  3. Hassani, S. Mathematical Physics: A Modern Introduction to Its Foundations. Springer, 2013. — physics convention (linearity in the second argument).
  4. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — scalar product / dot product.