The question is: Let $u$ be a unit length eigenvector of $Σ$ associated with the eigenvalue $λ_1$. Let $v$ be a unit length eigenvector of $Σ$ orthogonal to $u$ that is associated with the eigenvalue $λ_2$. Show that $Σ^{-1} = λ_{1}^{−1}uu^T + λ_{2}^{−1}vv^T$ holds. You need to show this for general positive definite symmetric Σ.
I expressed Σ as $ADA^{-1}$ where A is the matrix of eigenvectors and D of the eigenvectors. I'm not sure if this is the right way to approach this.
Any help would be great! Thanks
I assume $\Sigma$ is $2 \times 2$, otherwise the question does not make sense.
If $A$ has $u$ and $v$ as columns, and $D$ is diagonal with diagonal entries $\lambda_1$ and $\lambda_2$, then $\Sigma=ADA^{-1}$ (where $A^{-1} = A^\top$ since $A$ is orthogonal). Check that $\Sigma^{-1} = AD^{-1} A^{-1}$, and then rewrite this in terms of $\lambda_1, \lambda_2, u, v$.