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In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entriesall following an $a-$stable law divided by $N^{1/a}$. We prove that the leastsingular value of $D_{N}$, multiplied by $N$, tends to the same law as in theGaussian case, for almost all $a \in (0,2)$. This is proven by considering thesymmetrization of the matrix $D_{N}$ and using a version of the three stepstrategy, a well known strategy in the random matrix theory literature. Inorder to apply the three step strategy, we also prove an isotropic local lawfor the symmetrization of matrices after slightly perturbing them by a Gaussianmatrix with a similar structure. The isotropic local law is proven for ageneral class of matrices that satisfy some regularity assumption. We alsoprove the complete delocalization for the left and right singular vectors of$D_{N}$ at small energy, i.e., for energies at a small interval around $0$.

Universality of the least singular value and singular vector delocalisation for L\'evy non-symmetric random matrices