  We study the existence of the maximal quantum Fisher information matrix in
multi-parameter quantum estimation, which bounds the ultimate precision limit.
We show that when the maximal quantum Fisher information matrix exists, it can
be directly obtained from the underlying dynamics. Examples are then provided
to demonstrate the usefulness of the maximal quantum Fisher information matrix
by deriving various tradeoff relations in multi-parameter quantum estimation
and obtaining the bounds for the scalings of the precision limit.
