  A simple proof and detailed analysis on the non-ergodicity for
multidimensional harmonic oscillator systems with Nose-Hoover type thermostat
are given. The origin of the nonergodicity is symmetries in the
multidimensional target physical system, and is differ from that in the
Nose-Hoover thermostat with the 1-dimensional harmonic oscillator. A new simple
deterministic method to recover the ergodicity is also presented. An individual
thermostat variable is attached to each degree of freedom, and all these
variables act on a friction coefficient for each degree of freedom. This action
is linear and controlled by a Nos\'e mass matrix Q, which is a matrix analogue
of the scalar Nos\'e's mass. Matrix Q can break the symmetry and contribute to
attain the ergodicity.
