# Can we determine higher powers of a matrix in terms of lower powered matrices?

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?

• To be clear: Are you looking for a formula to generate the coefficients of the $A^k$ polynomial recursively from the coefficients of the $A^n$ polynomial? If so, have you tried a few steps to see what that recursion might look like?
– Blue
Commented Jun 12, 2021 at 6:35
• @Blue Yes, I am looking for a formula for that sort. I have tried to find the same but only for a handful of matrices. I wanted to know if there is a way to generalise. Commented Jun 12, 2021 at 8:58

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).