Let $A$ be a positive-definite matrix over the rings of quaternions with size $n\times n$. Let $f$ be a function that $f(A)=r_{1}^{m_{1}}...r_{t}^{m_{t}}$ where $r_i$'s are eigenvalues of $A$ and $ m_{i} $'s are algebraic multiplicities of $r_i$'s.
Is this true, $f(AB)=f(A)f(B)?