New characterizations and representations of the generalized Bott–Duffin inverse and its applications

Some new characterizations of the generalized Bott–Duffin inverse are proposed by using range, nullspace and matrix equations. We extend the famous fact that the rank equation characterizes the inverse of matrices to the generalized Bott–Duffin inverse, and give its several representations. Additionally, we show the relationships between the generalized Bott–Duffin inverse and two classes of nonsingular bordered matrices and apply them to the Cramer's rules of two classes of restricted linear equations.