Editing Dot product (section)

== Generalizations ==

=== Complex vectors ===
For vectors with [[complex number|complex]] entries, using the given definition of the dot product would lead to quite different properties. For instance, the dot product of a vector with itself could be zero without the vector being the zero vector (e.g. this would happen with the vector {{nowrap|<math>\mathbf{a} = [1\ i]</math>).}} This in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the dot product, through the alternative definition<ref>{{cite book | page = 287 | first= Sterling K. | last = Berberian | title = Linear Algebra | year = 2014 | orig-year = 1992 | publisher = Dover | isbn = 978-0-486-78055-9}}</ref><ref name="Lipschutz2009" />
<math display="block"> \mathbf{a} \cdot \mathbf{b} = \sum_i {{a_i}\,\overline{b_i}} ,</math>
where <math>\overline{b_i}</math> is the [[complex conjugate]] of <math>b_i</math>. When vectors are represented by [[column vector]]s, the dot product can be expressed as a [[matrix product]] involving a [[conjugate transpose]], denoted with the superscript H:
<math display="block"> \mathbf{a} \cdot \mathbf{b} = \mathbf{b}^\mathsf{H} \mathbf{a} .</math>

In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is [[sesquilinear]] rather than bilinear, as it is [[conjugate linear]] and not linear in <math>\mathbf{a}</math>. The dot product is not symmetric, since
<math display="block"> \mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}} .</math>
The angle between two complex vectors is then given by
<math display="block"> \cos \theta = \frac{\operatorname{Re} ( \mathbf{a} \cdot \mathbf{b} )}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } .</math>

The complex dot product leads to the notions of [[Hermitian form]]s and general [[inner product space]]s, which are widely used in mathematics and [[physics]].

{{anchor|Norm squared}}The self dot product of a complex vector <math>\mathbf{a} \cdot \mathbf{a} = \mathbf{a}^\mathsf{H} \mathbf{a} </math>, involving the conjugate transpose of a row vector, is also known as the '''norm squared''', <math display="inline">\mathbf{a} \cdot \mathbf{a} = \|\mathbf{a}\|^2</math>, after the [[Euclidean norm]]; it is a vector generalization of the ''[[absolute square]]'' of a complex scalar (see also: ''[[Squared Euclidean distance]]'').

=== Inner product ===
{{main|Inner product space}}
The inner product generalizes the dot product to [[vector space|abstract vector spaces]] over a [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]], being either the field of [[real number]]s <math> \R </math> or the field of [[complex number]]s <math> \Complex </math>. It is usually denoted using [[angular brackets]] by <math> \left\langle \mathbf{a} \, , \mathbf{b} \right\rangle </math>.

The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is [[Sesquilinear form|sesquilinear]] instead of bilinear. An inner product space is a [[normed vector space]], and the inner product of a vector with itself is real and positive-definite.

=== Functions ===
The dot product is defined for vectors that have a finite number of [[coordinate vector|entries]]. Thus these vectors can be regarded as [[discrete function]]s: a length-<math>n</math> vector <math>u</math> is, then, a function with [[domain of a function|domain]] <math>\{k\in\mathbb{N}:1\leq k \leq n\}</math>, and <math>u_i</math> is a notation for the image of <math>i</math> by the function/vector <math>u</math>.

This notion can be generalized to [[Square-integrable function|square-integrable functions]]:  just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some [[measure space]] <math>(X, \mathcal{A}, \mu)</math>:<ref name="Lipschutz2009" />
<math display="block"> \left\langle u , v \right\rangle = \int_X u v \, \text{d} \mu.</math>

For example, if <math> f</math> and <math>g</math> are [[Continuous function|continuous functions]] over a [[Compact space|compact subset]] <math> K</math> of <math>\mathbb{R}^n</math> with the standard [[Lebesgue measure]], the above definition becomes:
<math display="block"> \left\langle f , g \right\rangle = \int_K f(\mathbf{x}) g(\mathbf{x}) \, \operatorname{d}^n \mathbf{x} .</math>

Generalized further to [[complex function|complex continuous functions]] <math>\psi</math> and <math>\chi</math>, by analogy with the complex inner product above, gives:
<math display="block"> \left\langle \psi, \chi \right\rangle = \int_K \psi(z) \overline{\chi(z)} \, \text{d} z.</math>

=== Weight function ===
Inner products can have a [[weight function]] (i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions <math>u(x)</math> and <math>v(x)</math> with respect to the weight function <math>r(x)>0</math> is
<math display="block"> \left\langle u , v \right\rangle_r = \int_a^b r(x) u(x) v(x) \, d x.</math>

=== Dyadics and matrices ===
A double-dot product for [[Matrix (mathematics)|matrices]] is the [[Frobenius inner product]], which is analogous to the dot product on vectors. It is defined as the sum of the products of the corresponding components of two matrices <math>\mathbf{A}</math> and <math>\mathbf{B}</math> of the same size:
<math display="block"> \mathbf{A} : \mathbf{B} = \sum_i \sum_j A_{ij} \overline{B_{ij}} = \operatorname{tr} ( \mathbf{B}^\mathsf{H} \mathbf{A} ) = \operatorname{tr} ( \mathbf{A} \mathbf{B}^\mathsf{H} ) .</math>
And for real matrices,
<math display="block"> \mathbf{A} : \mathbf{B} = \sum_i \sum_j A_{ij} B_{ij} = \operatorname{tr} ( \mathbf{B}^\mathsf{T} \mathbf{A} ) = \operatorname{tr} ( \mathbf{A} \mathbf{B}^\mathsf{T} ) = \operatorname{tr} ( \mathbf{A}^\mathsf{T} \mathbf{B} ) = \operatorname{tr} ( \mathbf{B} \mathbf{A}^\mathsf{T} ) .</math>

Writing a matrix as a [[dyadics|dyadic]], we can define a different double-dot product (see ''{{slink|Dyadics|Product of dyadic and dyadic}}'') however it is not an inner product.

=== Tensors ===
The inner product between a [[tensor]] of order <math>n</math> and a tensor of order <math>m</math> is a tensor of order <math>n+m-2</math>, see ''[[Tensor contraction]]'' for details.