I am trying to solve the following optimization problem:
$$ \min_{x \in \mathbf{R}^2} \, x^T A x + b^Tx \quad \text{subject to $x^T J x = 1$} $$
where $A$ is a positive semi-definite $2 \times 2$ matrix and $J$ is a $2 \times 2$ diagonal matrix. I construct the Lagrangian $$ \mathcal{L} = x^T A x + b^Tx + \lambda(x^T J x - 1) $$ from which I obtain the first order condition $$ 2A x + b + 2 \lambda J x = 0 $$ I conclude that the solution must satisfy the expression $$ x = -\frac{1}{2}(A + \lambda J)^{-1}b $$ where $\lambda$ solves the equation $$ \frac{1}{4}b^T(A + \lambda J)^{-1}J(A + \lambda J)^{-1}b = 1 $$ This equation can be written as a fourth order polynomial in $\lambda$.
My question: How do I know that $A + \lambda J$ is nonsingular at the optimal value of $\lambda$?