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This is inspired by comments on another recent question on a matrix for the fourth-root of $X$. That question links to a paper of Muradian and Frias, who provide a number of nice equations and quantum gates for the nth root of a matrix $A$ by relating the matrix to a corresponding matrix-exponential of $A$, especially for self-inverse matrices of order two with $A^2=I$.

For example, Muradian and Frias provide the following equation for such matrices:

$$\sqrt[n]A=e^{i\frac{\pi}{2n}(I-A)},\:n=1,2,3,\ldots\tag 7$$

If we think of $A$ as an adjacency matrix of some large graph, then $I-A$ is close to a Laplacian matrix of the graph. This is interesting because the eigenvalues of the Laplacian matrix control the dynamics of continuous-time random walks on these graphs.

But how would that equation extend to other matrices that are not self-inverse, and instead have another lower order? For example, what would a similar equation be for a matrix $B$ of order four, e.g. $B^2\ne B^4=I$?

Is Muradian and Frias's equation (7) valid, or easily tweaked, for such matrices?

Mark Spinelli
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    I haven't worked it out in full, but it feels like observing that (under some assumptions on $A$) $$\log A = \log(I - (I-A)) = \sum_{k=0}^\infty \frac{1}{k}\sum_{j=0}^k \binom{k}{j} A^j=\sum_{\ell=0}^\infty A^\ell \sum_{k=0}^\infty \frac{1}{k+\ell} \binom{k+\ell}{\ell}$$ might be useful. Then using the minimal polynomial of $A$ one should be able to reduce this to an expression for the coefficients of each finite "useful" power of $A$. For example if $A^2=I$ one should get only two terms, and find a way to sum the arising series and obtain the result you stated. – glS May 02 '22 at 08:18

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