Editing Dot product (section)

=== 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]]'').