Approximation of a singular matrix by a non singular matrix and D-inverse

The authors discuss an approximation (with respect to Hilbert-Schmidt norm) of a singular matrix by a nonsingular matrix with a given determinant. They prove the existence and unicity of this matrix. The authors use this result to define the D-inverse of a matrix by requiring the determinant to be the product of the nonzero singular values of the given matrix. They discuss the interest of this approach for improving the conditioning of a given matrix.