I didn't find the exact definition (especially an explicit one). Let $k$ be a field, $B= \begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} $$\in M_2(k)$ a matrix. Let $k[B]$ be the algebra generated by $B$ in $M_2(k)$. Is there an explicit way to write $k[B]$?
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It is indeed the subalgebra of $M_2(k)$ generated by the powers of $B$. It is isomorphic to $k[x]/m(x)$, where $m(x)$ is the minimum polynomial of $B$. In your particular case $m(x) = \det(xI - B) = (x-a)(x-c)$ (unless a=c and b=0, in which case it's $x-a$).
Magdiragdag
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Just to complete the answer, and assuming $B \ne 0$, if $a-c=b=0$, then the algebra is just $\left{ \begin{bmatrix} u & 0\ 0 & u \end{bmatrix} :u \in k \right}$. Otherwise it is $\left{ \begin{bmatrix} u + v a & v b\ 0 & u + v c \end{bmatrix} :u, v \in k \right}$. – Andreas Caranti Feb 01 '13 at 10:35