pentadiagonal matrix

An $n\\times n$ pentadiagonal matrix (with $n\\geq 3$ ) isa matrix of the form

\\begin{pmatrix}c_{1}&d_{1}&e_{1}&0&\\cdots&\\cdots&0\\\\b_{1}&c_{2}&d_{2}&e_{2}&\\ddots&&\\vdots\\\\a_{1}&b_{2}&\\ddots&\\ddots&\\ddots&\\ddots&\\vdots\\\\0&a_{2}&\\ddots&\\ddots&\\ddots&e_{n-3}&0\\\\\\vdots&\\ddots&\\ddots&\\ddots&\\ddots&d_{n-2}&e_{n-2}\\\\\\vdots&&\\ddots&a_{n-3}&b_{n-2}&c_{n-1}&d_{n-1}\\\\0&\\cdots&\\cdots&0&a_{n-2}&b_{n-1}&c_{n}\\end{pmatrix}.

It follows that a pentadiagonal matrix is determined by five vectors:one $n$ -vector $c=(c_{1},\\ldots,c_{n})$ ,two $(n-1)$ -vectors $b=(b_{1},\\ldots,b_{n-1})$ and $d=(d_{1},\\ldots,d_{n-1})$ ,and two $(n-2)$ -vectors $a=(a_{1},\\ldots,a_{n-2})$ and $e=(e_{1},\\ldots,e_{n-2})$ .It follows that a pentadiagonal matrix is completely determinedby $n+2(n-1)+2(n-2)=5n-6$ scalars.