Pentadiagonal matrix

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

$\left(\begin{array}{ccccccc}{c}_1& d_1& e_1& 0& \cdots & \cdots & 0\\ b_1& c_2& d_2& e_2& \ddots & & ⋮\\ a_1& b_2& \ddots & \ddots & \ddots & \ddots & ⋮\\ 0& a_2& \ddots & \ddots & \ddots & e_{n-3}& 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & d_{n-2}& e_{n-2}\\ ⋮& & \ddots & a_{n-3}& b_{n-2}& c_{n-1}& d_{n-1}\\ 0& \cdots & \cdots & 0& a_{n-2}& b_{n-1}& c_n\end{array}\right)\in {\mathbb{ℝ}}^{n\times n}.$

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

See also

• Tridiagonal matrix
• Heptadiagonal matrix