Band Matrix - Band Storage

Band Storage

Band matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.

For example, a tridiagonal matrix has bandwidth 3. The 6-by-6 matrix

$\begin{bmatrix} B_{11} & B_{12} & 0 & \cdots & \cdots & 0 \\ B_{21} & B_{22} & B_{23} & \ddots & \ddots & \vdots \\ 0 & B_{32} & B_{33} & B_{34} & \ddots & \vdots \\ \vdots & \ddots & B_{43} & B_{44} & B_{45} & 0 \\ \vdots & \ddots & \ddots & B_{54} & B_{55} & B_{56} \\ 0 & \cdots & \cdots & 0 & B_{65} & B_{66} \end{bmatrix}$

is stored as the 6-by-3 matrix

$\begin{bmatrix} 0 & B_{11} & B_{12}\\ B_{21} & B_{22} & B_{23} \\ B_{32} & B_{33} & B_{34} \\ B_{43} & B_{44} & B_{45} \\ B_{54} & B_{55} & B_{56} \\ B_{65} & B_{66} & 0 \end{bmatrix}.$

A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6-by-6 matrix with a right bandwidth of 2:

$\begin{bmatrix} A_{11} & A_{12} & A_{13} & 0 & \cdots & 0 \\ & A_{22} & A_{23} & A_{24} & \ddots & \vdots \\ & & A_{33} & A_{34} & A_{35} & 0 \\ & & & A_{44} & A_{45} & A_{46} \\ & sym & & & A_{55} & A_{56} \\ & & & & & A_{66} \end{bmatrix}.$

This matrix is stored as the 6-by-3 matrix:

$\begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{22} & A_{23} & A_{24} \\ A_{33} & A_{34} & A_{35} \\ A_{44} & A_{45} & A_{46} \\ A_{55} & A_{56} & 0 \\ A_{66} & 0 & 0 \end{bmatrix}.$

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—Rutherford Birchard Hayes (1822–1893)

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—Henry David Thoreau (1817–1862)

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