One can see from the Cayley-Hamilton Theorem that for a $n\times n$ matrix, we can write any power of the matrix as a linear combination of lesser powers and the identity matrix, say if $A\neq cI_n$, $c\in \Bbb{C}$ is a given matrix, it can be written as a linear combination of $I_n,A^{-1},A,A^2,\cdots,A^{n-1}$. Is this representation unique? If so, is it under special cases? As an example let's consider this:
Let $A\neq cI_n$ , $c\in \Bbb{C}$ be a $3\times 3$ matrix over $\Bbb{C}$ and $A^3=k_3A^2+k_2A+k_1I_3=m_3A^2+m_2A+m_1I_3$ for $k_i,m_i\in \Bbb{C}$. Does it hold that $k_i=m_i \forall i=1,2,3?$
If we take a formalistic approach I guess we can say that the equivalence above consitutes a system of $3$ equations with $6$ variables ($k_i,m_i$).
That is, we evaluate the two linear combinations and by matrix equivalence we demand that all entries are equal: $a_{ij}=b_{ij}=c_{ij}$. But in general, unless the matrix $A$ has a "special" form, that will have infinite solutions.
Is this-overly simplistic-approach correct? Should I try to find counter examples perhaps or the question has some trivial answer that I overlook?