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

[math]\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\; \R^{n \times n} \,. }[/math]

It follows that a pentadiagonal matrix has at most [math]\displaystyle{ 5n-6 }[/math] 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