Matrix Representation of Inner Product

Definition

• With $V$ as a Vector Space
• Let $\beta =\{{\alpha }_1,{\alpha }_n\}$ as a Ordered Basis for $V$
• Suppose $⟨\cdot |\cdot ⟩$ is a Inner Product on $V$
• The matrix of the inner product is $G_{jk}:=(⟨a_k|a_j⟩)\in \mathbb{C}\text{ or }\mathbb{R}$

Examples

Find the matrix representation of an inner product ${\mathbb{R}}^n$ with a standard basis $\alpha =\{e_1,\dots ,e)n\}$

$$G=\left[\begin{array}{ccc}⟨e_1|e_1⟩& \dots & ⟨e_n|e_1⟩\\ ⋮& & \\ & & \\ ⟨e_1|e_n⟩& \dots & ⟨e_n|e_n⟩\end{array}\right]$$ \left[\begin{array}{cc} 1 & 0 \dots & 0\\ 0 & 1 \dots & 0\\ \vdots\\ 0 & 0 \dots & 1\\ \end{array}\right]$$