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Suppose $K_1$ and $K_2$ are positive definite n x n matrices. Suppose that for i = 1,2, the minimizer of $p_i(x) = x^TK_ix-2x^Tf_i + c_i$, is $x^*_i$. Is the minimizer of $p(x) = p_1(x) + p_2(x)$ given by $x^* = x^*_1 + x^*_2$? Prove or give a counterexample.

I know that for a positive definite matrix K and L we have that K + L is also positive definite. For this question, adding both i = 1,2 will provide a positive definite matrix, hence they both will have a minimum solution, but I don't know how to prove that adding the two minimization solution will also give the minimizer of the two? Is proving this similar to proving that two positive definite matrices will also provide a positive matrix so K + L > 0, or is this question false?

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    OK. Why not try constructing some simple $2\times2$ matrices, maybe even diagonal ones, to see what happens?2012-10-23

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This is false. Simply let $n=1$, $p_1=x^2$ and $p_2=x^2-2x+1$.