Heptadiagonal matrix

In linear algebra, a heptadiagonal matrix is a matrix that is nearly diagonal; to be exact, it is a matrix in which the only nonzero entries are on the main diagonal, and the first three diagonals above and below it. So it is of the form

[math]\displaystyle{ \begin{bmatrix} B_{11} & B_{12} & B_{13}& B_{14} & 0 & \cdots & 0 & 0 \\ B_{21} & B_{22} & B_{23} &B_{24}& B_{25} & 0 & \ddots & 0 \\ B_{31} & B_{32} & B_{33} & B_{34} & B_{35} & B_{36} & \ddots & \vdots \\ B_{41} & B_{42} & B_{43} & B_{44} & B_{45} & B_{46}& B_{47} & 0 \\ 0 & B_{52}& B_{53} & B_{54} & B_{55} & B_{56}& B_{57}& B_{58} \\ \vdots & \ddots & B_{63}& B_{64} & B_{65} & B_{66} & B_{67}& B_{68} \\ 0 & \cdots & 0 & B_{74}& B_{75} & B_{76} & B_{77} & B_{78} \\ 0 & 0 & \cdots & 0 & B_{85} & B_{86} & B_{87} & B_{88} \end{bmatrix} }[/math]

It follows that a heptadiagonal matrix has at most [math]\displaystyle{ 7n-12 }[/math] nonzero entries, where n is the size of the matrix. Hence, heptadiagonal matrices are sparse. This makes them useful in numerical analysis.

See also

• Tridiagonal matrix
• Pentadiagonal matrix