= Solution
For a <linear transformation> $T$ of a <finite-dimensional vector space> over a <field> $F$, its <minimal polynomial> $m_T$ is the unique <monic polynomial> of least degree satisfying $m_T(T)=0$. There is at least one nonzero <polynomial> with this property because the powers $I,T,T^2,\ldots$ lie in the finite-dimensional <vector space> of linear maps $V\to V$. Dividing any other such <polynomial> by $m_T$ leaves a remainder of smaller degree that also vanishes at $T$, so that remainder must be zero. This also proves uniqueness.
If $C$ represents $T$ in any <basis>, the paper's <characteristic polynomial> is
$$
\boxed{\chi_T(x)=\det(C-xI).}
$$
It is independent of the <basis>, since changing the <basis> replaces $C-xI$ by a <similar matrix>, preserving its <determinant>. Its degree is $\dim V$ and its leading coefficient is $(-1)^{\dim V}$; $m_T$ is always <monic polynomial>[monic].
Back to article page
