= Characteristic and minimal polynomials of right multiplication
{title2=$\chi_{\rho_A}=\chi_A^2,\quad m_{\rho_A}=m_A$}
On two-by-two complex matrices let $\rho_A(X)=XA$. In row-entry coordinates, its matrix is $\operatorname{diag}(A^T,A^T)$, so its <characteristic polynomial> is the square of that of $A$. For every polynomial $p$, $p(\rho_A)(X)=Xp(A)$. Thus $p(A)=0$ implies $p(\rho_A)=0$, and the reverse implication follows by taking $X=I$. The operators have exactly the same annihilating polynomials and hence the same <minimal polynomial>. In general, right multiplication on $r$-by-$n$ matrices has characteristic polynomial $\chi_A^r$ and the same minimal polynomial.
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