 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015/13/a/ii/solution

Repeated substitution in the affine iteration gives

x_{k} = M^{k - 1} x_{1} + (I + M + \dots + M^{k - 2}) c .

Therefore the required matrix polynomials are

f (M) = M^{k - 1}, g (M) = j = 0 \sum k - 2 M^{j} .

These formulas do not require an inverse. If

I - M

is an invertible matrix, multiplying the finite geometric sum by

I - M

gives

g (M) = (I - M^{k - 1}) (I - M)^{- 1}

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