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Suppose we are given a list of $N$ positive definite quadratic forms $X^TQ_k X$ (where $k\in[1,N]$ and $Q_k\in\mathbb{R}^{p\times p}$ $\forall k$), and a positive vector $V$ of same length $N$ i.e. $V=(v_k>0)_{k\in[1,N]}$.

My question is the following: how do we find the vector $X^*\in\mathbb{R}^p$ minimising the squared distance between the array of quadratic forms and vector $V$, i.e. solving: $$X^*=\underset{X}{\mathrm{argmin}} \sum_{k=1}^N \left(X^TQ_k X - v_k\right)^2 \quad ?$$ The scalar case ($p$=1) is easy to solve, giving $X^*=\pm\sqrt{\frac{\sum q_kv_k}{\sum q_k^2}}$, but I'm not sure how to proceed in the general case $p$>1.

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  • $\begingroup$ Writing the minimisation problem as a zero first-order derivative in $X$ when $p>1$, we end up with the following equation for $X^*$: $$\sum_{k=1}^N ({X^*}^T Q_k X^*-v_k)\times Q_kX^*=0.$$ Yet not clue how to solve this, whether analytically or numerically. Does this correspond to a known type of equation ? $\endgroup$ Commented Feb 14 at 14:51
  • $\begingroup$ I suggest you tell us what happens when $p=2, N=2.$ Or do such numerical examples with, say, everything integers $\endgroup$ Commented Feb 14 at 19:05
  • $\begingroup$ Integers ? Also N=2 I'm afraid doesn't teach much. To clarify: the requirement for me is to find generic solutions workable in the $p\sim500$ and $N\sim 100$ regime. $\endgroup$ Commented Feb 14 at 23:30

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