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I'm interested in computing of the product $M^n v$, where $M$ is an $m\times m$ matrix (over a semiring) and $v$ is column-vector, with the smallest number of multiplications in the underlying semiring and in the regime of $n=\Theta(m)$.

If we first compute $M^n$ using fast exponentiation, this would take $\Theta(\log n)$ matrix multiplications, totaling in $\Theta(m^3\log n)$ semiring multiplications (assuming naive matrix multiplication).

On the other hand, if we just iteratively compute $M^k v$ for $k=1,2,\dots,n$, each time multiplying a vector by $M$, then it'd take us $\Theta(m^2n)$ semiring multiplications.

So, fast exponentiation appears to be useless. Is there any other way to speed up things here?

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    $\begingroup$ Is $M$ a diagonalizable matrix? Or maybe can be converted to Jordan normal form? In that case, this might help you. $\endgroup$
    – nir shahar
    Commented Jan 20, 2022 at 22:31
  • $\begingroup$ @nirshahar: Diagonalizable matrices exist over fields, in my case matrix is over a semiring, which in fact contains no invertible elements except the multiplicative identity. $\endgroup$ Commented Jan 20, 2022 at 23:40

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