  This work delves into the exponential time differencing (ETD) schemes for the
matrix-valued Allen-Cahn equation. In fact, the maximum bound principle (MBP)
for the first- and second-order ETD schemes is presented in a prior publication
[SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field.
Noteworthy is our novel contribution, demonstrating that the first- and
second-order ETD schemes for the matrix-valued Allen-Cahn equation -- both
being linear schemes -- unconditionally preserve the MBP, even in instances of
nonsymmetric initial conditions. Additionally, we prove that these two ETD
schemes preserve the energy dissipation law unconditionally for the
matrix-valued Allen-Cahn equation. Some numerical examples are presented to
verify our theoretical results and to simulate the evolution of corresponding
matrix fields.
