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I have a set of square matrices $\{A_i\}_{i \in \{1,..., n\}}$ and another square matrix of equal size $K$. Under the assumption that such a matrix exists and is unique, I want to find the unique $B$ that satisfies the following equation:

$$\sum_{i=1}^n(A_i+B)^{-1}=K$$

All matrices are of full rank and symmetric. Does anyone know of a way, algebraically or computationally, for me to work out the value of $B$?

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  • $\begingroup$ I'm not sure I understand - what happened to the inverse terms in the product? $\endgroup$
    – JDoe2
    Jul 31 '20 at 0:58
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This does not seem a simple matrix equation to solve. Computationally, my first attempt would be with Newton's method, even if it takes $O(k^6)$ per iteration, where $k$ is the size of the matrices. The Jacobian of the map in the LHS is $$ L_B f[H] = \sum_{i=1}^n (A_i+B)^{-1}H(A_i+B)^{-1}, $$ and to solve the equation $L_Bf[H] = Y$ for $H$ given $Y$ you need to convert it to a $k^2 \times k^2$ linear system (there are faster algorithms to solve this linear matrix equation for $n=2$, but I do not think there is anything better otherwise).

If you need to solve it for dimensions for which this is unfeasible, then I would try turning it into a fixed-point equation, for instance $$ B = \left(K - \sum_{i=2}^n (A_i+B)^{-1}\right)^{-1} - A_1, $$ and then hope that the iteration $$ B_{m+1} = \left(K - \sum_{i=2}^n (A_i+B_m)^{-1}\right)^{-1} - A_1, $$ converges.

The scalar version of this equation is a secular equation, but searching for this term I found nothing interesting for matrix arguments.

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  • $\begingroup$ Hmm... I see - just to clarify - I am a little unfamiliar with fixed-point equations. How would I be able to prove convergence for a fixed point equation? $\endgroup$
    – JDoe2
    Jul 26 '20 at 6:26
  • $\begingroup$ @JDoe2 The typical strategy is showing that the right-hand side $g(B) = (K-\sum_{i>1} (A_i+B)^{-1})^{-1}-A_1$ is a contraction, but that does not seem obvious to me. Another possible way is showing that $B_{m+1}\geq B_m$ in a certain ordering, for instance the positive-definite ordering or the componentwise ordering, but that requires some hypotheses on your matrices. $\endgroup$ Jul 26 '20 at 17:22
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    $\begingroup$ In practice, I suggest to implement it first, and then if it converges you can worry about proving theoretical properties. $\endgroup$ Jul 26 '20 at 17:23
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If $n\geq 2$, generically, you are unlikely to encounter such an equation with only one solution.

Let $d$ be the dimension of our symmetric matrices.

$\bullet$ Consider the case $n=2$. Then the algebraic resolution of the system leads to obtaining the roots of a real polynomial $P$ of degree $2^d$ (generically). Our system admits a unique solution iff $P$ admits only one real root. Since $degree(P)$ is even, this is not possible.

Then we must assume that $n\geq 3$.

$\bullet$ Note that a random polynomial of degree $p$ has

$O(\log(p))$ real zeros on average -when the coefficients are independent standard normals-

or $O(\sqrt{p})$ -when the variance varies with the index of the coefficient-.

If $A_1,A_2,K$ are randomly chosen, then $P$ can be roughly considered as random and the probability that $P$ admits a single real root is very small when $p$ is large.

$\bullet$ For example, if $d=2,n=3$, we obtain -generically- a polynomial of degree $11$. I did some tests and got $5,7,9$ or $11$ real roots.

Finally, the existence of a unique solution is a very special case.

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