In linear algebra, a pentadiagonal matrix is a special case of band matrices. Its only nonzero entries are on the main diagonal and the first two upper and two lower diagonals. So, it is of the form.

${\displaystyle {\begin{pmatrix}c_{1}&d_{1}&e_{1}&0&\cdots &\cdots &0\\b_{1}&c_{2}&d_{2}&e_{2}&\ddots &&\vdots \\a_{1}&b_{2}&\ddots &\ddots &\ddots &\ddots &\vdots \\0&a_{2}&\ddots &\ddots &\ddots &e_{n-3}&0\\\vdots &\ddots &\ddots &\ddots &\ddots &d_{n-2}&e_{n-2}\\\vdots &&\ddots &a_{n-3}&b_{n-2}&c_{n-1}&d_{n-1}\\0&\cdots &\cdots &0&a_{n-2}&b_{n-1}&c_{n}\end{pmatrix))\in \;\mathbb {R} ^{n\times n}\,.}$

It follows that a pentadiagonal matrix has at most ${\displaystyle 5n-6}$ nonzero entries, where n is the size of the matrix. Hence, pentadiagonal matrices are sparse, making them useful in numerical analysis.