# Proving inner product features

On the space $\mathbb{C}^n$ we define the scalar product: $\langle x,y\rangle = \sum_{i=1}^n x_i y ̅_i$ , now we define the operator: $[\space ,\space ]: \mathbb{C}^n\times \mathbb{C}^n\to\mathbb{C}$ as $[x,y]= \langle Ax,y\rangle$ when $A$ is an invertible complex matrix.

1. Which condition should be required for $A$ to have complex conjugate : $[y,x]= \overline{[x,y]}$ ?

2. Assuming that $A$ satisfies the condition of Section 1, which condition should be required for $A$ to have positivity: $[x,x]=0\iff x=0,[x,x]\geq 0$?

For finding these conditions, I need to use the inner product rules:

(a) $\langle x,y\rangle =\overline{\langle y,x\rangle}$ (the bar denotes the complex conjugate)

(b) $\langle αx +βy, z\rangle =α\langle x, z\rangle + β\langle y, z\rangle$

(c) $\langle x,x\rangle \geq 0$

(d) $\langle x,x\rangle = 0$ implies x = 0.

I've already managed to prove linearity of the product easily.

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