For vectors with complex components, the real dot product must be modified so that a vector has a real nonnegative squared length. The standard complex inner product is $$\langle \mathbf u,\mathbf v\rangle=\sum_i u_i^v_i,$$ where $u_i^$ is the complex conjugate of $u_i$.

The order matters through conjugation: $$\langle \mathbf v,\mathbf u\rangle=\langle \mathbf u,\mathbf v\rangle^*.$$ In particular, $$\langle \mathbf v,\mathbf v\rangle=\sum_i|v_i|^2\ge0,$$ and it vanishes only for the zero vector. The norm is $$|\mathbf v|=\sqrt{\langle\mathbf v,\mathbf v\rangle}.$$

For a matrix $A$, the conjugate transpose or adjoint matrix is $$A^\dagger=(A^*)^{\mathsf T}.$$ It is the operation satisfying $$\langle \mathbf u,A\mathbf v\rangle =\langle A^\dagger\mathbf u,\mathbf v\rangle.$$

For $$A=\begin{pmatrix}1&i\2&3-i\end{pmatrix},$$ $$A^\dagger= \begin{pmatrix}1&2\-i&3+i\end{pmatrix}.$$

Two complex vectors are orthogonal when their inner product is zero. An orthonormal basis satisfies $$\langle e_i,e_j\rangle=\delta_{ij}.$$

Complex conjugation is essential: using an ordinary transpose would allow nonzero vectors to have zero or even complex 'squared length'. Complex inner products extend Euclidean geometry to the vector spaces used for waves, Fourier analysis and quantum states.