Hessenberg matrix

Remarks

Example: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & 8 \end{pmatrix}$ is upper Hessenberg, because its only entry with $i > j + 1$ is $a_{31} = 0$. Non-example: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ is not upper Hessenberg, because $a_{31} = 7 \ne 0$. Every upper triangular matrix is upper Hessenberg. A matrix that is both upper and lower Hessenberg is a tridiagonal matrix. An upper Hessenberg matrix is unreduced when all its subdiagonal entries $a_{i+1, i}$ are nonzero.