Let \(A^*\) denote the conjugate transpose of a matrix \(A\) and let \(\mathbb F\) be either \(\mathbb R\) or \(\mathbb C\). Dissipative Hamiltonian matrix pencils are pencils of the form \[ P(\lambda)=\lambda E-A=\lambda E-(J-R)Q,\tag{1} \] where \(E,J,R\in\mathbb F^{n,n}\) satisfy the conditions \(J^*=-J\) and the matrices \(Q^*E=E^*Q\), \(R^*=R\) are positive semidefinite. The authors prove that an arbitrary matrix pencil \(L(\lambda)\in\mathbb F^{n,n}[\lambda]\) is strictly equivalent to a pencil of form (1) if and only of the following conditions hold:

1.

the spectrum of \(L(\lambda)\) is contained in the closed left half plane;

2.

the finite nonzero eigenvalues on the imaginary axis are semisimple, and the partial multiplicities of the eigenvalue zero are at most two;

3.

the index of \(L(\lambda)\) is at most two;

4.

the left minimal indices are all zero, and the right minimal indices are at most one.

They introduce the concept of a posH pencils, i.e., pencils of the form \[ \lambda(J_1+R_1)+(J_2+R_2), \] where \(J_1=-J_1^*\), \(J_2=-J_2^*\), and \(R_1\), \(R_2\) are positive semidefinite. Such pencils arise as linearizations of matrix polynomials with positive definite or semidefinite Hermitian coefficients. The authors characterize the possible Kronecker structures for posH matrix pencils. The obtained criterion implies some necessary or sufficient conditions for the regularity of such pencils. The numerical range of posH pencils is also studied.
The last part of the paper is devoted to the matrix polynomials \(P(\lambda)=\sum_{i=0}^d\lambda^iA_i\) whose skew-Hermitian parts of the coefficients are equal to zero. The authors prove that if for even \(d\) the matrix \(A_0\) is invertible, then the index of \(P(\lambda)\) does not exceed \(d\). They also localize the spectrum of \(P(\lambda)\) for the case when \(d=3\), \(A_1\) is positive semidefinite, and \(A_0\), \(A_2\), \(A_3\), \(A_2+A_1\) are positive definite.