Hessenberg matrix

\begin{bmatrix}a_{11}&a_{12}&a_{13}&\cdots&a_{1,n-1}&a_{1n}\\ a_{21}&a_{22}&a_{23}&\cdots&a_{2,n-1}&a_{2n}\\ 0&a_{32}&a_{33}&\cdots&a_{3,n-1}&a_{3n}\\ 0&0&a_{43}&\cdots&a_{4,n-1}&a_{4n}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&a_{n,n-1}&a_{nn}\end{bmatrix}

and a lower Hessenberg matrix is of the form

\begin{bmatrix}a_{11}&a_{12}&0&\cdots&0&0\\ a_{21}&a_{22}&a_{23}&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ a_{n-2,1}&a_{n-2,2}&a_{n-2,3}&\cdots&a_{n-2,n-1}&0\\ a_{n-1,1}&a_{n-1,2}&a_{n-1,3}&\cdots&a_{n-1,n-1}&a_{n-1,n}\\ a_{n,1}&a_{n,2}&a_{n,3}&\cdots&a_{n,n-1}&a_{n,n}\end{bmatrix}