Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks

We study the eigenvalue bounds for the nonsingular saddle point matrices of Hermitian and indefinite (1,1)(1,1) and (2,2)(2,2) blocks without imposing the restrictions that the (1,1)(1,1) blocks are positive definite on the kernels of the (2,1)(2,1) blocks.