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As stated in the title, the matrix equation is as follows:

$$ AB^n = C $$

where $A$ and $C$ are known matrices of size $1 \times m$.

$$ \begin{bmatrix} a_1 & a_2 & \dots & a_m \end{bmatrix}B^n= \begin{bmatrix} c_1 & c_2 & \dots & c_m \end{bmatrix} $$

The power $n$ is also known. The problem is to find any matrix $B$ of size $m \times m$ which solves this equation. Are there any algorithms or numerical methods that can successfully tackle this problem?

Rodrigo de Azevedo
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MrHowever
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  • Find any invertible matrix D such that $D^T A^T = C^T$, then solve $B^n = D$ using https://math.stackexchange.com/questions/646919/nth-root-of-a-matrix – 10762409 Jun 28 '23 at 16:58
  • What is the underlying field? $\mathbb R$, $\mathbb C$ or something else? – user1551 Jun 28 '23 at 20:13
  • @10762409 Doesn't this work only when D has positive eigenvalues? When a negative eigenvalue appears, the logarithm becomes non-principal, and the exponentiation doesn't result in the original matrix (at least according to Matlab). – MrHowever Jun 29 '23 at 00:39
  • @user1551 Technically, in the original equation, all the values are integers. However, I don't think this is solvable with integer arithmetic, so let's say they are real numbers. – MrHowever Jun 29 '23 at 00:42

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