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Generate successive matrix powers and take cumulative sums using matrix addition:For the stipulated square matrix, iterate (multMat m) m produces m, m*m, m*(m*m), and so on. Associativity of exact matrix multiplication identifies these as $M,M^2,M^3,\dots$. The left scan scanl1 addMat then returnsAll dimensions remain compatible. The infinite outer list is available a finite prefix at a time under lazy evaluation; no convergence of the matrix geometric series is needed. With floating-point entries, the expression specifies this particular multiplication and addition order, since machine arithmetic need not obey the exact algebraic laws.

The most general type signature is scanl :: (b -> a -> b) -> b -> [a] -> [b]. The left fold step consumes an accumulator of type b and an input of type a, returning the next accumulator. A left scan retains all these accumulator values, including the initial one.

Use map k (u:us) = k u : map k us, list-map fusion, and the left fold equation

foldl f e (x:ys) = foldl f (f e x) ys

For the empty list, the prefix definition immediately gives scanl f e [] = [e]. For a nonempty list, calculate:

scanl f e (x:xs)
  = map (foldl f e) ([] : map (x:) (inits xs))
  = e : map ((foldl f e) . (x:)) (inits xs)
  = e : map (foldl f (f e x)) (inits xs)
  = e : scanl f (f e x) xs

The second-to-third line uses the displayed fold equation as an equality of functions of ys. Hence the efficient left scan is:

scanl :: (b -> a -> b) -> b -> [a] -> [b]
scanl f e [] = [e]
scanl f e (x:xs) = e : scanl f (f e x) xs

On a finite input of length $n$, fully demanding the original prefix-fold results performs $0+1+\cdots +n=n(n+1)/2$ accumulator updates. The recurrence performs $n$. Thus the update count improves from $\Theta (n^2)$ to $\Theta (n)$, assuming constant-cost updates. Both return $n+1$ results. This does not assert constant space for arbitrary accumulators or imply that lazy accumulators are evaluated strictly.