  Quaternion symmetry is ubiquitous in the physical sciences. As such, much
work has been afforded over the years to the development of efficient schemes
to exploit this symmetry using real and complex linear algebra. Recent years
have also seen many advances in the formal theoretical development of
explicitly quaternion linear algebra with promising applications in image
processing and machine learning. Despite these advances, there do not currently
exist optimized software implementations of quaternion linear algebra. The
leverage of optimized linear algebra software is crucial in the achievement of
high levels of performance on modern computing architectures, and thus provides
a central tool in the development of high-performance scientific software. In
this work, a case will be made for the efficacy of high-performance quaternion
linear algebra software for appropriate problems. In this pursuit, an optimized
software implementation of quaternion matrix multiplication will be presented
and will be shown to outperform a vendor tuned implementation for the analogous
complex matrix operation. The results of this work pave the path for further
development of high-performance quaternion linear algebra software which will
improve the performance of the next generation of applicable scientific
applications.
