Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 1 / 1H (Linear Algebra)

The minimal polynomial $m_A$ is the monic polynomial of smallest degree satisfying $m_A(A)=0$. An annihilating polynomial exists, for example because $I,A,\dots ,A^{n^2}$ are linearly dependent in the $n^2$-dimensional space of matrices. A nonzero constant cannot annihilate $A$, so the minimal degree is positive.

If $p$ and $q$ are two such monic polynomials of minimal degree $d$, then $(p-q)(A)=0$ and $\mathrm{deg}(p-q)<d$ unless $p=q$. Minimality therefore forces $p=q$. Polynomial division also shows that $m_A$ divides every annihilating polynomial: the remainder after division would otherwise be an annihilator of smaller degree. This proves uniqueness rather than merely choosing one of several minimal-degree polynomials.

If $A$ is real, coefficientwise complex conjugation of $m_A(A)=0$ gives $\overline{m_A}(A)=0$. The conjugated polynomial is monic of the same degree, hence uniqueness gives $\overline{m_A}=m_A$. The minimal polynomial of a real matrix has real coefficients.

For $n\ge 3$, use the block diagonal matrixThe two-dimensional Jordan block is annihilated by $(t-1{)}^2$ but not by $t-1$, while the scalar block requires the factor $t+1$. A polynomial annihilates the direct sum exactly when it annihilates every block. Thus its minimal polynomial is the least common multiple $(t-1{)}^2(t+1)$. For $n=3$ the final identity block is omitted.