Inner Product on Vector Space

Question

How are we supposed to define inner products on give vector space?

For example, how can we define 3 different inner products on $\mathbb{F}$$^n$ where $\mathbb{F}$ $\in$ {$\mathbb{R}$, $\mathbb{C}$}?

Answer 1

One easy way to construct an inner product $(\cdot,\cdot)$ is to define $$ (u,v) = \langle Au, Av \rangle $$ where $\langle \cdot,\cdot \rangle$ denotes the usual inner product (i.e. the "dot-product") and $A$ is any invertible matrix. As it turns out, every inner product over $\Bbb R^n$ and $\Bbb C^n$ can be constructed in this way.

Answer 2

Every symmetric, bilinear, positive definite and non-degenerate form on $\mathbb{R}^n \times \mathbb{R}^n$ gives a scalar product. In $\mathbb{C}^n \times \mathbb{C}^n$ you require anti-linearity instead. Therefore, any $n \times n$ matrix with elements in $\mathbb{R}^n$ or $\mathbb{C}^n$ and the above properties gives an inner product.

For example, since the standard scalar product in $\mathbb{R}^3$ is given by the identity matrix, three different inner products are (trivially) given by $$ \left( \begin{matrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{matrix} \right) \quad \left( \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} \right) \quad \left( \begin{matrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{matrix} \right). $$

Answer 3

Let $A_1(v,w) $ be one inner product, e.g. $A_1(v,w) = \sum_i v_i\overline{w_i}$ then you can define an infinite family of inner products by $A_\lambda = \lambda A_1 $ for $\lambda > 0 $.