Are linear combinations of powers of a matrix unique?

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?

2 Answers

The situation is as follows: let $$d$$ be the degree of the minimal polynomial of $$A$$. Then the matrices $$\{I,A,A^2,\dots,A^{d-1}\}$$ are linearly independent and form a basis for the span of the powers of $$A$$.

In particular, this means that $$A$$ will have the uniqueness property you describe if and only if the minimal and characteristic polynomials coincide. That is, the following are equivalent:

1. The matrices $$\{I,A,\dots, A^{n-1}\}$$ are linearly independent
2. $$\Bbb R^n$$ is $$A$$-cyclic
3. $$A$$ is similar to a companion matrix
4. $$A$$ is non-derogatory
5. The minimal polynomial of $$A$$ is the same as its characteristic polynomial
6. The Jordan form of $$A$$ has one Jordan block for each eigenvalue
7. All eigenspaces of $$A$$ have dimension at most $$1$$
8. A matrix $$B$$ satisfies $$AB = BA$$ if and only if there is a polynomial $$f(x)$$ for which $$B = f(A)$$.

So, for example: any $$A$$ with $$n$$ distinct eigenvalues will have this property, and any $$A$$ which consists of a single Jordan block will have this property. Any diagonalizable $$A$$ with a repeated eigenvalue will not have this property.

• Thank you very much for your thorough and informative answer! – MathematicianByMistake May 16 '16 at 16:45

Take $A=2I$ the identity matrix, $A^3=8I =2A^2$ so the coefficients are not unique.

Take $A$ nilpotent and not zero $A^3=0$, you have $A^3=0A^2=0A$.

• Thanks yet again! I will make sure that no multiples of $I_n$ are considered valid-otherwise we have trivial counterexamples. – MathematicianByMistake May 16 '16 at 16:36