This problem comes from solving nonlinear least squares with sparse Levenberg-Marquardt algorithm, to derive a iteration increment $\Delta$ we have to solve following system: $$(J^TJ+\lambda \operatorname{diag}(J^TJ))\Delta=-J^T\epsilon$$ where $\operatorname{diag}(J^TJ)=\operatorname{diag}(j_1,j_2,j_3,....)$ is a diagonal matrix with diagonal elements corresponding to those of $J^TJ$ in turn, $\epsilon$ is the optimization object to be minimized.
Now write $x^T(J^TJ+\lambda \operatorname{diag}(J^TJ))x$ as $\|Jx\|^2+\sum_ij_ix_i^2$, it seems not unnecessarily positive even if $j_i \ge0$ for symmetric positive semi-definite matrix $J^TJ$
You are right, the matrix $M=J^TJ+\lambda\, \mathrm{diag}(J^TJ)$ on the left side of your displayed equation need not be invertible: if, as for instance when $J$ is the zero matrix or more generally when one of the $j_i = 0$. But you really want to solve the equation $M\Delta=-J^T\epsilon$, which can be solved, since the range of $M$ is the column space of $J^T$.
Suppose $j_1=0$, for example. Then the first entry in the right hand $-J^T\epsilon$ will equal $0$, as will all the entries in the first row and column of $M$. That it, the coefficient of $\Delta_1$ in your equation system is $0$, and you can reduce your equation set by dropping that variable. And so on.