# System of matrix equations

Problem definition: Let $$x_i \in \mathbb{R}^d$$ and $$a_i \in [0,1]$$, for all $$i = 1,\dots, k$$ (with $$k\geq d$$). Define $$M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$$ and assume $$M(a) \succ 0.$$

Question: Is there any closed-form solution (for $$a$$) to this set of equations? If not, there is at least an efficient way to solve it?

$$\begin{cases}\text{trace}\left(M(a)^{-1}x_jx_j^T\right) = \text{trace}\left(M(a)^{-1}x_lx_l^T\right), \forall j \neq l,\\ \sum_{i =1}^k a_i= 1.\end{cases}$$

Related question: here.

Attempted solution: Note that, since $$\text{trace}(Ayx^T) = \text{trace}(x^TAy) = x^TAy$$, the first equation can be rewritten as

$$x_j^T M(a)^{-1}x_j =x_l^T M(a)^{-1}x_l,$$

with $$M(a) = X^TAX$$, where $$X: \text{col}(X) = \{x_i\}_{i\in[k]}$$, and $$A = \text{diag}(a)$$.

Hence we have $$(x_j^TX^{-T})A^{-1}(X^{-1}x_j) = (x_l^TX^{-T})A^{-1}(X^{-1}x_l)$$

By denoting $$\tilde{x}_i = x_i^TX^{-T}$$, for all $$i\in[k]$$, and for a vector $$x$$, its $$n$$-th component $$x(n)$$, we can write

$$\sum_{i = 1}^k \frac{\tilde{x_j}(i)^2}{a_i} = \sum_{i = 1}^k \frac{\tilde{x_l}(i)^2}{a_i}.$$

Any suggestion on the correctness of this solution would be appreciated.

• simplest case (for reference): $d=1$, $a_i=x_i^2/(\sum_j x_j^2)$. Commented Apr 5, 2021 at 17:26
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