linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

, 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. :

\begin 
   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_ & 0 \\
   \vdots & \ddots & \ddots & \ddots & \ddots & d_ & e_ \\
   \vdots & & \ddots & a_ & b_ & c_ & d_ \\
   0 & \cdots & \cdots & 0 & a_ & b_ & c_n 
\end
\in\; \R^
\,.

It follows that a pentadiagonal matrix has at most

5n-6

nonzero entries, where ''n'' is the size of the

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

. Hence, pentadiagonal matrices are sparse, making them useful in

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

See also