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pd = positive definite

I'm having a tough time grappling with the concept of pd, psd matrices in general. My understanding basically just boils down to this, basically after multiplying everything out using the matrix formula x'Ax, you will get some sort of polynomial. If you get a polynomial where everything is squared, you're in good shape because it's impossible for the equation to be negative.

A question that I have is, what does knowing that a matrix A is psd or pd tell us about the entries of A? Doesn't it only tell us how the entries of A interact (ie, when they are added up, you get something >= 0). If that's true, how would adding the identity matrix give you anymore information about the definiteness of the matrix?

Thanks!