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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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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many would consider your post rude because it is just the problem statement, not a request for help, so consider rewriting it. –  Zev Chonoles Feb 10 '13 at 20:11
    
Also, you can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. –  Zev Chonoles Feb 10 '13 at 20:11
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""Revisable""...complex matrix? What's a "revisable" matrix? –  DonAntonio Feb 10 '13 at 20:11
    
I’ve removed the (homework) tag: it should never be added without explicit acknowledgement by the OP that the question is homework. –  Brian M. Scott Feb 11 '13 at 7:37
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