We have already met the characteristic polynomial, \(p_T(x)\), of a linear operator \(T\) on a vector space \(V\), which, when the underlying field is algebraically closed, factors as \(p_T(x)=\prod_{i=1}^r(x-\lambda_i)^{n_i}\) with \(n_i\) the dimension of the generalised eigenspace of the eigenvalue \(\lambda_i\). It is an example of an annihilating polynomial of \(T\), that is, a polynomial \(f\) such that \(f(T)=0\). The minimal polynomial of \(T\) is the annihilating polynomial of \(T\) of least degree.

Under the assumption of an algebraically closed field, we know that \(T\) may be represented by a matrix \(\mathbf{T}\) in Jordan normal form. For a given eigenvalue, \(\lambda_i\), let us denote by \(k_i\) the maximal order of the Jordan blocks associated with \(\lambda_i\). Then the minimal polynomial of \(T\) is given by
\begin{equation}
m_T(x)=\prod_{i=1}^r(x-\lambda_i)^{k_i}.
\end{equation}

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