  In this paper, we first study the projections onto the set of unit dual
quaternions, and the set of dual quaternion vectors with unit norms. Then we
propose a power method for computing the dominant eigenvalue of a dual
quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the
sequence generated by the power method converges to the dominant eigenvalue and
its corresponding eigenvector linearly. For a general dominant eigenvalue, we
show the standard part of the sequence generated by the power method converges
to the standard part of the dominant eigenvalue and its corresponding
eigenvector linearly. Based upon these, we reformulate the simultaneous
localization and mapping (SLAM) problem as a rank-one dual quaternion
completion problem. A two-block coordinate descent method is proposed to solve
this problem. One block has a closed-form solution and the other block is the
best rank-one approximation problem of a dual quaternion Hermitian matrix,
which can be computed by the power method. Numerical experiments are presented
to show the efficiency of our proposed power method.
