Editing Sparse matrix (section)

===Banded===
{{main article|Band matrix}}
An important special type of sparse matrices is [[band matrix]], defined as follows. The [[lower bandwidth of a matrix]] {{math|'''A'''}} is the smallest number {{math|''p''}} such that the entry {{math|''a''<sub>''i'',''j''</sub>}} vanishes whenever {{math|''i'' > ''j'' + ''p''}}. Similarly, the [[Band matrix#upper bandwidth|upper bandwidth]] is the smallest number {{math|''p''}} such that {{math|1=''a''<sub>''i'',''j''</sub> = 0}} whenever {{math|''i'' < ''j'' − ''p''}} {{harv|Golub|Van Loan|1996|loc=§1.2.1}}. For example, a [[tridiagonal matrix]] has lower bandwidth {{math|1}} and upper bandwidth {{math|1}}. As another example, the following sparse matrix has lower and upper bandwidth both equal to 3. Notice that zeros are represented with dots for clarity.
<math display="block">\begin{bmatrix}
  X   &   X   &   X   & \cdot & \cdot & \cdot & \cdot & \\
  X   &   X   & \cdot &   X   &   X   & \cdot & \cdot & \\
  X   & \cdot &   X   & \cdot &   X   & \cdot & \cdot & \\
\cdot &   X   & \cdot &   X   & \cdot &   X   & \cdot & \\
\cdot &   X   &   X   & \cdot &   X   &   X   &   X   & \\
\cdot & \cdot & \cdot &   X   &   X   &   X   & \cdot & \\      
\cdot & \cdot & \cdot & \cdot &   X   & \cdot &   X   & \\              
\end{bmatrix}</math>

Matrices with reasonably small upper and lower bandwidth are known as band matrices and often lend themselves to simpler algorithms than general sparse matrices; or one can sometimes apply dense matrix algorithms and gain efficiency simply by looping over a reduced number of indices.

By rearranging the rows and columns of a matrix {{math|'''A'''}} it may be possible to obtain a matrix {{math|'''A'''′}} with a lower bandwidth. A number of algorithms are designed for [[Graph bandwidth|bandwidth minimization]].

====Diagonal====
A very efficient structure for an extreme case of band matrices, the ''[[diagonal matrix]]'', is to store just the entries in the [[main diagonal]] as a [[one-dimensional array]], so a diagonal {{math|''n'' × ''n''}} matrix requires only {{math|''n''}} entries.