 Differentiation on an exponential-polynomial space

For distinct real or complex numbers ${\lambda }_j$ and positive integers $m_j$, the vector space spanned by $t^k{e}^{{\lambda }_{j}t}$, $0\le k<m_j$, is preserved by differentiation. In each ordered basis $(e^{{\lambda }_{j}t},t{e}^{{\lambda }_{j}t},\dots )$, its matrix has ${\lambda }_j$ on the diagonal and $1,2,\dots ,m_j-1$ just above it. This follows from the product rule. Rescaling to $t^k{e}^{{\lambda }_{j}t}/k!$ makes each block a Jordan block. In particular, differentiation is invertible on this vector space exactly when every ${\lambda }_j\ne 0$: the determinant is ${\prod }_j{\lambda }_j^{m_j}$.