SAMS Subject Classification Number: 11

Consider an \(n \times n\) matrix \(B\) in the indefinite inner product space generated by an invertible Hermitian matrix \(H\). The matrix \(B\) is called \(H\)-nonnegative if the indefinite inner product \([Bx,x]\) is nonnegative for all \(x\in\mathbf{C}^n\). Then \(B\) has a simple structure as \(B\) is also \(H\)-selfadjoint, \(B\) has only real eigenvalues and the Jordan normal form of \(B\) is almost diagonal since all of the Jordan blocks have size \(1\) except for the Jordan blocks corresponding to the zero eigenvalue which have size at most \(2\).

We will take a look at conditions for the existence of square roots (with and without structure) of \(H\)-nonnegative matrices as well as the description of the roots in the nilpotent case.