A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems. (English) Zbl 0936.65045

Computation of eigenvalues of a \((2n,2n)\) skew-Hamiltonian matrix \(N\) with complex elements is effected by transforming the \((4n,4n)\) block diagonal matrix with diagonal blocks \(N\), \(\overline N\) into a real skew-Hamiltonian matrix \({\mathcal N}\) whose eigenvalues agree with those of \(N\). The matrix \({\mathcal N}\) may be put into a skew-Hamiltonian Schur from \({\mathcal R}\) and the eigenvalues may be obtained by a method of C. F. Van Loan [Linear Algebra Appl. 61, 233-251 (1984; Zbl 0565.65018)]. The results apply to complex Hamiltonian matrices as well since if \(N\) is skew-Hamiltonian then \(H= (-iN)\) is Hamiltonian.
The eigenspaces associated with the positive and negative eigenvalues of \(H\), \(\text{Inv}_+H\), \(\text{Inv}_-H\), are computed from the Hamiltonian Schur form \((-i{\mathcal R})\) assuming that \(H\) has no purely imaginary eigenvalues. These eigenspaces are useful in certain control problems. A code ZHAEV has been written using the new numerical method and an error analysis of the method is presented. In two numerical tests ZHAEV is compared with ZGEEV, a general code for determining eigenvalues of nonsymmetric matrices. The results obtained show that the new code is faster and more accurate than the more general code.


65F15 Numerical computation of eigenvalues and eigenvectors of matrices
93B36 \(H^\infty\)-control
93B40 Computational methods in systems theory (MSC2010)


Zbl 0565.65018


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