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For $m\geq n$, Let $K$ be an algebraic closed base field, and define$\tau_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kerneldimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, andis usually called determinantal variety. In most cases $\tau_{m,n,k}$ issingular with singular locus $\tau_{m,n,k+1}$. In this paper we compute thelocal Euler obstruction of $\tau_{m,n,k}$, and we prove that the characteristiccycle of the intersection cohomology complex of $\tau_{m,n,k}$ is irreducible.We also give an explicit formula for the Chern-Mather class of $\tau_{m,n,k}$as a class in projective space. The irreducibility of the intersectioncohomology characteristic cycle follows from the explicit computation of thelocal Euler obstruction, a study of the `Tjurina transforms' of determinantalvarieties, and the Kashiwara-Dubson's microlocal index theorem.  Our explicit formulas are based on calculations of degrees of certain Chernclasses of the universal bundles over the Grassmannian. We use The Schubert 2package in Macaulay2 to exhibit examples of the Chern-Mather class and theclass of the characteristic cycle of $\tau_{m,n,k}$ for some small values of$m,n,k$.  Over the complex numbers, the local Euler obstruction of $\tau_{m,n,k}$ wasrecently computed by N.~Grulha, T.~Gaffney and M.~Ruas by methods in complexgeometry.

Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties