inner product with orthonormal basis

Topics

• vector inner product

Let $u_1,\dots ,u_n$ be an orthonormal (orthogonal and normal aka length 1) basis of $V$.

$$\begin{array}{rlr}& \text{If,}& \mathbf{x}=\sum _{k=1}^n{x}_k{\mathbf{u}}_k,\mathbf{y}=\sum _{k=1}^n{y}_k{\mathbf{u}}_k\\ & \text{then,}& ⟨\mathbf{x},\mathbf{y}⟩=\sum _{k=1}^n\overline{x_k}{y}_k\end{array}$$

where $\overline{x}$ represents the complex conjugate of $x$. In the context of inner products:

• The conjugate ensures the inner product satisfies $⟨x,y⟩=\overline{⟨y,x⟩}$
• For real numbers, $\overline{x}=x$, and $⟨\mathbf{x},\mathbf{y}⟩={\mathbf{x}}^T\mathbf{y}$

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