I am learning a bit more about CAPM, and wanted to know if there was a specific way that weightings of assets in the optimal mean-variance portfolio changed (for constant risk aversion, expected return, and risk free rate) if the covariance matrix was altered in different ways.
For example, for $n$ risky assets there is an $n \times n$ covariance matrix $\Sigma$ which holds the covariances of the assets in each of its indices. So I was wondering what would happen to the portfolio weights if 'suddenly' the covariance matrix was changed such that risky assets all became independent but their variances did not change. This would make a new covariance matrix, $\Sigma'$, a diagonal matrix, with the same diagonal as $\Sigma$.
So I know the weights of the risky assets should be $w = \lambda \Sigma^{-1}(\mu - r \mathbb 1)$. In our case let's just make the risk aversion, $\lambda = 1$, and the risk free rate $r = 0$. So the formula for each individual weight, $w_i, i \in [1,n]$ should be $w_i = \sum_{k = 1}^{n} \Sigma^{-1}_{ik}\cdot\mu_k$.
So in the case where we use the diagonal covariance matrix, $\Sigma'$, we would have that $w_i = \frac{\mu_i}{\text{Var}_i}$, and now I need to compare $\sum_{k = 1}^{n} \Sigma^{-1}_{ik}\cdot\mu_k$ and $\frac{\mu_i}{\text{Var}_i}$, but since the inverse of the covariance matrix is in the first equation, I cannot tell whether the coefficients of each $\mu_k$ will be lead to values less than or greater than $\frac{\mu_i}{\text{Var}_i}$.
So I was wondering if there is a definite answer to my question (i.e. do the weights necessarily change in a certain manner), or if it depends on the particular covariance matrix of the risky assets.