AN ANALYTICAL METHOD TO CALCULATE THE ERGODIC AND DIFFERENCE MATRICES OF THE DISCOUNTED MARKOV DECISION PROCESSES



and

Djki, k = 1, 2, 3, . . . , N, are constant factors different from zero.

Proof:

From linear algebra results, that the determinant of matrix (I - βP ) always exist,
it is bigger than zero and it is a polynomial of
N - degree. Each polynomial of N -
degree has exactly
N different real roots. Hence we can show det (I - βP ) in form
(16). Let prove (15) and write this formula in the following form:

[Dji (β )]  =  [⅞]  + jd⅛l ...JDNL

(17)


det (I βP)   (1 — α1β)   (1 — α2β)        (1 — αNβ)

This formula shows decomposition of the left side of dependence (17) into sum
of
N partial fractions. Such a decomposition is always possible. If we want to cal-
culate values of factors
Djki, k = 1, 2, 3, . . . , N, we should solve (N × N) systems
of equations:

Dji (β)
det (I βP)


D1      D2.           Dn          ___

(18)


--j— +--j— +  +--j-—  ij = 1 N
1 — αι β +1 — α2β +   +1 — αN β ,j ,

it means

Dji (β)    = Dji (1 — α2β) ... (1 — αNβ) + Dji (1 — α1β) (1 — α3β) . . .

det (I βP)                       det (I βP)

... (1 — αNβ) + ... + DjN (1 — αιβ) (1 — α2β)... (1 — αN-ιβ)
det (I βP)

After rejection of denominators of both sides we compare factors which stand in
front of the same powers
β, β0, β1 , . . . , βN-1 of the left and right side of nume-
rators.

So each element of matrix

[Dji (β)]   = Γ   Dji (β)

i,j = 1,N


det (I βP) det (I βP)



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