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I came across the following matrix while reading an article..Can you please help me to understand the following.

We are defining following form: $$\sum_{i,j}\partial_{z_i}\partial_{\bar{z_j}}rdz_i\otimes d\bar{z_j}$$

Where $r: M(\subset \mathbb C^2) \to \mathbb R$ is $C^2$ function.

1- Where this form acts[Domain]; I think $dz_i\otimes d\bar{z_i}$ should act on $\mathbb C\times \mathbb C$. But how does it act.. etc.

2- What does mean by eigen value of this form, what is typical eigen vector..

i am sorry if question are trivial... but can someone help me.

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Hi Junu, what is your background with tensor products? en.wikipedia.org/wiki/Tensor_product –  tentaclenorm May 6 '12 at 4:51
    
@tentaclenorm, I know that $dz_i\otimes d\bar{z_i}$ should act on $\mathbb C\times \mathbb C$. But how does it act.. etc.. i think, i know a bit of tensor.. but i am confused.. –  Junu May 6 '12 at 5:03

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