= Cube root of a matrix
{title2=$B^3=A$}
= Cube roots of a matrix
{synonym}
A <cube root of a matrix> $A$ is a <square matrix> $B$ of the same size over the specified <field> satisfying $B^3=A$. Every complex <invertible matrix> has one: put it in <Jordan normal form>, choose a scalar <cube root> of each nonzero <eigenvalue>, and apply the <truncated binomial cube root of the identity plus a nilpotent matrix> on each block. The <Nilpotent Jordan block> $J_3(0)$ has none, even though its square is nonzero. If $B^3=J_3(0)$, then $B^9=0$, making $B$ a <nilpotent matrix> of size three; the <Cayley-Hamilton theorem> then gives $B^3=0$, a contradiction.
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