Eigenvalues of Dual Hermitian Matrices with Application in Formation Control
arxiv(2024)
摘要
We propose a supplement matrix method for computing eigenvalues of a dual
Hermitian matrix, and discuss its application in multi-agent formation control.
Suppose we have a ring, which can be the real field, the complex field, or the
quaternion ring. We study dual number symmetric matrices, dual complex
Hermitian matrices and dual quaternion Hermitian matrices in a unified frame of
dual Hermitian matrices. An n × n dual Hermitian matrix has n dual
number eigenvalues. We define determinant, characteristic polynomial and
supplement matrices for a dual Hermitian matrix. Supplement matrices are
Hermitian matrices in the original ring. The standard parts of the eigenvalues
of that dual Hermitian matrix are the eigenvalues of the standard part
Hermitian matrix in the original ring, while the dual parts of the eigenvalues
of that dual Hermitian matrix are the eigenvalues of those supplement matrices.
Hence, by applying any practical method for computing eigenvalues of Hermitian
matrices in the original ring, we have a practical method for computing
eigenvalues of a dual Hermitian matrix. We call this method the supplement
matrix method. In multi-agent formation control, a desired relative
configuration scheme may be given. People need to know if this scheme is
reasonable such that a feasible solution of configurations of these
multi-agents exists. By exploring the eigenvalue problem of dual Hermitian
matrices, and its link with the unit gain graph theory, we open a
cross-disciplinary approach to solve the relative configuration problem.
Numerical experiments are reported.
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