Why is this method valid to show function is PSD?

Master1022
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Homework Statement
We are looking at the function: ## f(x, y) = \frac{x^2}{y}, x \in \mathbb{R} , y > 0 ##. Is this function positive semi-definite.
Relevant Equations
Hessian
Hi,

I was looking at the following example, and I cannot really understand the justification for why this function is positive semi-definite.

Example Problem: We are looking at the function: ## f(x, y) = \frac{x^2}{y}, x \in \mathbb{R} , y > 0 ##. Is this function positive semi-definite.

Solution provided: We can calculate the Hessian to yield:
\nabla^2 f(x, y) = \frac{2}{y^3} \begin{pmatrix} y^2 & -xy \\ -xy & x^2 \end{pmatrix} = \frac{2}{y^3} \begin{pmatrix} y \\ -x \end{pmatrix} \begin{pmatrix} y \\ -x \end{pmatrix}^T \succcurlyeq 0

Therefore, the function is PSD.

My Question: I am not fully convinced by the last step of the argument. I can understand the decomposition, but it isn't obvious to me why that implied positive semi-definiteness.

From what I understand, a matrix ##A## can be said to be PSD if ## x^T Ax \geq 0## for all ##x##. We can also check that the eigenvalues all have real parts ## \geq 0 ##. However, the solution above shows that the Hessian matrix is separable and is multiplied by a pre-factor which is always ## \geq 0 ## because ## y > 0##. I mean, perhaps I could 'transpose' the expression they have and think about the ## \frac{2}{y^3} ## as the ## A ## matrix, but I am also not sure that is valid to do.Any help would be greatly appreciated as I am sure this is quite a simple question and I am missing something obvious.
 
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Here is an exercise:

For any column vector ##v##, show that ##vv^T## is a positive semi definite matrix.
 
Orodruin said:
Here is an exercise:

For any column vector ##v##, show that ##vv^T## is a positive semi definite matrix.
Ah yes.

Okay so the ## v v^T ## resembles a covariance matrix. It has rank 1 because row can be written as ## v_i v ## (that is element ## v_i ## multiplied by ## v ##) and so we can do some row reduction. Therefore because it is rank 1, it has 1 non-zero eigenvalue. We know that the eigenvalues of a matrix sum to the trace so the one non-zero eigenvalue is ## \lambda = ||v||_2 ^2 ##, while the other ## n - 1 ## eigenvalues are 0.

We can see that the eigenvalues are ## \geq 0 ## and therefore ## v v^T ## is PSD.

Relating it back to my question, we have a ## k v v^T ## with some positive constant ##k## and so now that makes sense why the matrix is PSD.

Thank you.
 
While that works, the more straightforward way is to directly use the definition of positive semi definite. For any ##u##, ##a = u^T v## is a number and so ##a = u^T v = v^T u##. Hence, with ##A = vv^T##, ##u^TAu = u^T v v^T u = a^2 \geq 0## and thus ##vv^T## is positive semi definite.
 
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