Given a matrix $\Omega$ of size $m \times m$ that is not positive semi definite (some of the eigenvalues are negative) but symmetrical, I want to simulate $\pmb{Z} \sim N_{(m)}(\pmb{0}, \Omega)$. I tried first with the SVD such that $$ \Omega = U S V^T$$ but given that $\Omega$ has negative eigenvalues, I get that $U \neq V$, so setting $$\pmb{Z} = U S^{1/2}\pmb{X} $$ where $\pmb{X} \sim N_{(m)}(\pmb{0}, I_n)$ does not work. I also tried with the spectral decomposition, getting $$ \Omega = Q \Lambda Q^{-1} $$ but I cannot get $\Lambda^{1/2}$ without using complex numbers, and I am not sure that using a complex normal vector is appropriate. I have also attempted using certain algorithms that try to get a positive semidefinite matrix as close to $\Omega$ as possible (using Frobenius norm), but the matrix is not as close as expected. Finally, I also tried using $$ D = \frac{1}{2}(\Omega + \Omega^T) $$ and with $m = min(\text{eigen}(D))$ where $\text{eigen}(D)$ are the eigenvalues of $D$, establish $$ \Omega^\prime = D +(\lambda-m)I_n $$ and use $\Omega^\prime$ instead of $\Omega$, which successfully recovers the covariances, but the diagonal is overestimated by a factor of 4x, which is expected given that the procedure only ensures that the outer-diagonal elements are preserved.
My question is, ¿is this possible? And if it is, ¿how?
EDIT:
If it is not possible, ¿why not? Maybe that could give me a hint on how to proceed.