I am studying positive semi-definite matrices. A natural problem arises when we study the properties of the corresponding quadratic form of such matrices: Can we always represent such quadratic forms as sums of squares (of course, of linear terms)?
I am aware that a high-degree, non-negative polynomial need not be a sum of squares (the $6$-th degree Motzkin’s Example shows that). But if we are restricting this to only second-degree polynomials, does this still hold true?
I really appreciate any insight/hint for my question.