Let \(k\) be a field and \(n\geq 1\). A matrix \(A\in\Mat_{n\times n}(k)\) is nilpotent if there exists an integer \(m\geq 1\) such that \(A^m = 0\).
A strictly upper triangular matrix \(A\in\Mat_{n\times n}(k)\) is nilpotent, with \(A^n=0\): if \(e_1,\ldots,e_n\) is the standard basis, then \(A\) maps \(\spanof\theset{e_1,\ldots,e_j}\) into \(\spanof\theset{e_1,\ldots,e_{j-1}}\) for each \(j\). Conversely, every nilpotent \(A\in\Mat_{n\times n}(k)\) is similar to a strictly upper triangular matrix: every eigenvalue \(\lambda\) of \(A\) in an algebraic closure of \(k\) satisfies \(\lambda^m=0\), so the characteristic polynomial of \(A\) is \(\pm x^n\), which splits over \(k\), and a matrix whose characteristic polynomial splits is similar to an upper triangular matrix whose diagonal entries are its eigenvalues.