Search

This work focuses on the class of $ \mathcal {L} $ L -positive semidefinite matrices A such that $ AA_{(\mathcal {L})}^{(\dag)}-A_{(\mathcal {L})}^{(\dag)}A $ A A (L) (†) − A (L) (†) A is nonsingular, where $ A_{(\mathcal {L})}^{(\dag)} $ A (L) (†) is the generalized Bott-Duffin inverse of A with respect to a subspace $ \mathcal {L} $ L . We give some equivalent characterizations for the nonsingularity of $ AA_{(\mathcal {L})}^{(\dag)}-A_{(\mathcal {L})}^{(\dag)}A $ A A (L) (†) − A (L) (†) A , $ AA_{(\mathcal {L})}^{(\dag)}+A_{(\mathcal {L})}^{(\dag)}A $ A A (L) (†) + A (L) (†) A and $ I_{n}-A(A_{(\mathcal {L})}^{(\dag)})^2A $ I n − A (A (L) (†) ) 2 A mainly by rank equalities and subspace operations, among which relations are pointed out. Meanwhile we show the inverses of their general versions in a decomposition form. In addition, the continuity of this class of matrices is discussed based on the continuity of the generalized Bott–Duffin inverse. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]