Hessenberg matrix

An upper Hessenberg matrix is of the form

\\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}