quality:corruption
Complex vectors

For vectors with 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 a = {\displaystyle \mathbf {a} =}). 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 a ⋅ b = ∑ i a i b i ¯, {\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i}{{a_{i}}\,{\overline {b_{i}}}},} where b i ¯ {\displaystyle {\overline {b_{i}}}} is the complex conjugate of b i {\displaystyle b_{i}}. When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H:

a ⋅ b = b H a. {\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} ^{\mathsf {H}}\mathbf {a}.}

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 a {\displaystyle \mathbf {a} }. The dot product is not symmetric, since a ⋅ b = b ⋅ a ¯. {\displaystyle \mathbf {a} \cdot \mathbf {b} ={\overline {\mathbf {b} \cdot \mathbf {a} }}.} The angle between two complex vectors is then given by cos θ = Re (a ⋅ b) ‖ a ‖ ‖ b ‖. {\displaystyle \cos \theta ={\frac {\operatorname {Re} (\mathbf {a} \cdot \mathbf {b})}{\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}}.}

The complex dot product leads to the notions of Hermitian forms and general inner product spaces, which are widely used in mathematics and physics.

The self dot product of a complex vector a ⋅ a = a H a {\displaystyle \mathbf {a} \cdot \mathbf {a} =\mathbf {a} ^{\mathsf {H}}\mathbf {a} }, involving the conjugate transpose of a row vector, is also known as the norm squared, a ⋅ a = ‖ a ‖ 2 {\textstyle \mathbf {a} \cdot \mathbf {a} =\|\mathbf {a} \|^{2}}, after the Euclidean norm; it is a vector generalization of the absolute square of a complex scalar (see also: squared Euclidean distance).