Engineering Rock Mechanics

Symmetry of interaction matrices^23 1

'hb

Transition probability matrix

for Markov chain of state changes

3

p --p --p ----x p., U =I

Asymmetric because of

direction dependence

Figure 14.8 Transition probability matrix for Markov chain of state changes.

probability of it remaining in that state or moving to one of the other states
is given by the transition probabilities shown in the matrix. If the parameter
is in state A, the probability that it will remain in that state is Puu; the
probability that it will move to state B is Pub; and the probability that it will
move to state C is Pa,. These probabilities are given in the first row of the
matrix, and their sum is unity. The second and third rows of the matrix
represent similar transition probabilities for parameters in states B and C.
Such transition probability matrices are used for generating Markov chains
of events and for studying the sensitivity of the occurrence of certain states
as a function of the transition probabilities. One could, for example,
consider in a geological analysis the types of sedimentary sequences that
will occur as the result of different depositional states.
The matrix illustrates symmetry and asymmetry conditions well. It may
be that Pub = P, or that Pub f Pb,. Quite clearly, any asymmetry of the matrix
results from directional dependence in the state change.
Another excellent example of asymmetry is the formulae for the
transformation of axes. In Fig. 14.9, the new co-ordinates x' and y' are given
as a function of the old co-ordinates x and y and the angle 8 through which
the axes have been rotated. It can be seen that the basic operation of
rotating the axes produces a cos 8 term along the leading diagonal
(representing the primary operation), and a sin^8 term in the off-diagonal
positions (representing the interaction between the axes). However, the
matrix is skew-symmetric because of the rotational nature of the
transformation: if we were to rotate the axes in the opposite direction, the
sign of the off-diagonal terms would be reversed. It should be noted that
this axis interaction is directly analogous to the note in Fig. 5.4, where
simple shear also involves an interaction between the axes.
Finally, in Fig. 14.10, the axes are rotated through 45", 90" and 180". For
the first case, the primary operation terms on the leading diagonal and the
interaction terms in the off-diagonal positions have equal importance, but
the matrix is still asymmetric. In the second case, the primary operation has
been reduced to zero, because a rotation of 90" can be considered simply
as an interchange of the axes with the signs of the off-diagonal
components indicating the positive directions of the new axes relative to