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

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

It follows that a heptadiagonal matrix has at most $7n-12$ 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