Consider a n-ordered square matrix A. Using Cayley-Hamilton Theorem, I can represent the matrix $A^n$ as a matrix polynomial P(A) of degree n-1.
Further any matrix $A^k$ where $k>n$ can also be represented as follows:
$$A^k= a_{k,n-1} A^{n-1} + a_{k,n-2} A^{n-2}+a_{k,n-3} A^{n-3} ... + a_{k,2} A^{2}+a_{k,1} A^{1} + a_{k,0}I$$
What I want to know that is there any way to determine the coefficients as a function of k?
You could find a representation using polynomial division by the characteristic polynomial.
For example, if $A = \begin{bmatrix} 0&1&0\\-1&0&0\\0&0&1\end{bmatrix}$ then $A$ has characteristic polynomial $x^{3}-x^{2}+x-1$. Then if you wanted to write $A^{4}$ you could use the fact that $$x^{4} = (x+1)(x^{3}-x^{2}+x-1) +1$$ to write $$A^{4} = (A+I)(A^{3}-A^{2}+A-I) +I = I.$$
The representation will not in general be unique as a polynomial of degree less than $n$; for a unique representation, you would want to use the minimal polynomial instead of the characteristic polynomial (which would let you write $A^{k}$ uniquely as a polynomial in $A$ of degree at most $d-1$, where $d$ is the degree of the minimal polynomial).
