Complex vector inner products

In order to preserve the property that the inner product of any vector with itself is the magnitude of that vector squared, we need to define the inner product of complex vectors so that

${\vec {\mathbf {u}}}\bullet {\vec {\mathbf {v}}}=\sum _{k=1}^{n}u_{k}^{*}v_{k}$

where we have assumed that the basis vectors are orthonormal and that there are $n$ dimensions. The $u_{k}^{*}$ is the complex conjugate of $u_{k}$. Some define the conjugate on the second vector instead of the first, but in order to be consistent with bra-ket notation from quantum mechanics, we do it with the conjugate on the first vector.

With this we have

${\vec {\mathbf {u}}}\bullet {\vec {\mathbf {v}}}=({\vec {\mathbf {v}}}\bullet {\vec {\mathbf {u}}})^{*}$

and

${\vec {\mathbf {v}}}\bullet {\vec {\mathbf {v}}}=|{\vec {\mathbf {v}}}|^{2}$.

Orthogonal functions

Principle author of this page: Rob Frohne