Upper triangular matrix

An upper triangular matrix has zero entries below its main diagonal. Its determinant is the product of its diagonal entries, and an invertible upper triangular linear system is solved by backward substitution.

A square matrix is strictly upper triangular if every entry on or below its diagonal is zero. A product of $n$ such $n$-by-$n$ matrices vanishes, so they form a nilpotent Lie algebra under the commutator. Multiplication by an upper triangular matrix preserves strict upper triangularity and zero trace.