Defining functions for manifolds of matrices

We consider defining systems for manifolds of possibly nonsquare matrices. The functions are algebraic expressions in the entries of the matrix. The systems are local in the sense of being defined on the sets of a predefined cover of the manifold with open and dense sets. We use bordered matrix techniques to characterize the manifold of matrices with a given rank. In the case of square matrices we give conditions to express that zero is an eigenvalue with a given algebraic multiplicity (and geometric multiplicity 1). In both cases the obtained systems have full linear rank; hence they define the underlying manifolds in a regular way and can be used analytically to obtain local parametrizations of the manifold or numerically to set up Newton systems with local quadratic convergence.