  In this paper, the quaternion matrix UTV (QUTV) decomposition and quaternion
tensor UTV (QTUTV) decomposition are proposed. To begin, the terms QUTV and
QTUTV are defined, followed by the algorithms. Subsequently, by employing
random sampling from the quaternion normal distribution, randomized QUTV and
randomized QTUTV are generated to provide enhanced algorithmic efficiency.
These techniques produce decompositions that are straightforward 9 to
understand and require minimal cost. Furthermore, theoretical analysis is
discussed. Specifically, the upper bounds for approximating QUTV on the rank-K
and QTUTV on the TQt-rank K errors are provided, followed by deterministic
error bounds and average-case error bounds for the randomized situations, which
demonstrate the correlation between the accuracy of the low-rank approximation
and the singular values. Finally, numerous numerical experiments are presented
to verify that the proposed algorithms work more efficiently and with similar
relative errors compared to other comparable decomposition methods. For the
novel decompositions, the theory analysis offers a solid theoretical basis and
the experiments show significant potential for the associated processing tasks
of color images and color videos.
