= Centralizer of a nonscalar two by two matrix
{title2=$AB=BA\iff B=\alpha I+\beta A$}
For any real nonscalar $2\times2$ <matrix> $A$, its commuting matrices are exactly $B=\alpha I+\beta A$ with $\alpha,\beta\in\mathbb R$. There is a vector $v$ for which $v,Av$ form a <basis>: otherwise every nonzero vector would be an <eigenvector>, and applying this to two independent vectors and their sum would force $A$ to be scalar. Given a commuting $B$, uniquely write $Bv=\alpha v+\beta Av$. Then $BAv=ABv=(\alpha I+\beta A)Av$, so $B$ and $\alpha I+\beta A$ agree on a basis and are equal. Conversely every polynomial in $A$ commutes with it. This generalizes the <centralizer of a planar quarter-turn>; the scalar-matrix exception is essential, because every matrix commutes with a scalar matrix.
Back to article page
