How to solve for $X - X^{-1} = W$ given $X$ is symmetric matrix

Question

I was trying to solve an optimization problem and the problem is simplified to solve

$X - X^{-1} = W$ and $X$ is symmetric matrix.

I was given the hint that I can perform eigenvalue decompoisiton on $W$, and construct a diagonal solution to obtain $X$, but not sure how to implement it as I'm not good at linear algebra.

Any idea or thoughts on this?

Answer 1

If $(v_i, \lambda_i)$ is an eigenpair of $X$, then $$X v_i - X^{-1} v_i = W v_i\Rightarrow Wv_i = (\lambda_i - \lambda_i^{-1}) v_i, $$ Therefore, $(v_i, \lambda_i - \lambda_i^{-1})$ is an eigenpair of $W$. So you can solve the eigendecomposition of $W$ first.

For each eigenvalue $\mu_i$ of $W$, solve $\lambda_ i - \lambda_i^{-1} = \mu_i$, thus $\lambda_i = \frac{1}{2}(\mu_i \pm \sqrt{\mu_i^2 + 4})$. Note there are two solutions for each, not sure if your problem has additional requirements, e.g. positive definiteness of $X$?

For instance, in MATLAB

[V, D] = eig(W);
mu = diag(D);
lambda = 0.5 * (mu + sqrt(mu.^2 + 4)); % assume positive definiteness of X
DX = diag(lambda);
X = V * DX * V';