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?

  • $\begingroup$ 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? $\endgroup$
    – Blue
    Jun 12, 2021 at 6:35
  • $\begingroup$ @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. $\endgroup$
    – Tony Stark
    Jun 12, 2021 at 8:58

1 Answer 1


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).


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