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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.

RobPratt
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ElementX
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    Yes, you can always do a linear change of variable to make it a signed sum of squares. If some square has negative sign, you can make all other squares zero and that square negative. See, one algorithm here – NDB Jul 19 '23 at 11:13
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    Cholesky-decompose the matrix – Rodrigo de Azevedo Jul 19 '23 at 22:21

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