Consider the problem of projecting a symmetric matrix $A$ onto the PSD cone. If $A$ can be written as the sum of eg $2$ other symmetric matrices, is the projection of $A$ the sum of individual projections (as would be the case with Euclidean projection eg on a linear subspace)? If this is not true in general, are there special cases where this holds (eg if $A = B + C$ and $B$ is PSD but $C$ isn't)?
2026-03-30 11:22:42.1774869762
Is Euclidean projection onto the PSD a linear operation?
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I suppose this depends on how exactly you define the projection of a vector (read: matrix) onto a cone, but for example if you take A a PSD matrix then presumably 2A projects onto itself while -A projects onto the zero matrix, however 2A + (-A) = A projects onto itself as well. Therefore the sum of the projections is 2A whereas the projection of the sum would just be A itself. The hypothesis does not hold.