represents the sequence of positions on the diagonal band that starts with {i,j} in a sparse array.


represents the positions between {imin,jmin,} and {imax,jmax,}.


represents positions starting with {imin,jmin,} and then moving with step {di,dj,}.


  • Band[pos]->v represents values v that repeat along the diagonal band starting at pos.
  • Band[pos]->{v1,v2,} represents a sequence of values vn along the diagonal band starting at pos.
  • With an array a of the same rank as the whole sparse array, Band[start]->a by default inserts a at the position specified by start.
  • With arrays an of the same rank as the whole sparse array, Band[pos]->{a1,a2,} represents a sequence of non-overlapping subarrays.
  • Band[start] is effectively equivalent to Band[start,{-1,-1,}].
  • For scalar values, Band[start,end] is equivalent to Band[start,end,{1,1,}].
  • Band[start,end,step] stops when any coordinate first exceeds its value in end.
  • Band[start,end]->{v1,v2,} takes the values vn to repeat cyclically until end is reached.
  • Band[start,Automatic,step]->a continues until the edge of the array is reached.
  • Band[start,Automatic,step]->{v1,v2,} continues until the vi are exhausted.


open allclose all

Basic Examples  (1)

Create a band diagonal matrix:

Convert to normal lists:

Scope  (11)

Mix Band with other SparseArray element specifications:

Give explicit values to fill in on the band:

Repeat the values cyclically:

Start the band at any position in the matrix:

Specify any start and end locations:

Step by 2 between elements on the band:

Any step can be used:

Automatically continue the band to the edge of the array:

Specify a band that is part of a row:

Specify an anti-diagonal matrix:

Insert a submatrix beginning at position 3, 3:

Cyclically repeat the submatrix:

Alternate the submatrix with a single element:

Generalizations & Extensions  (2)

Band works in sparse arrays of any rank:

Fill in a plane of values into a 3D sparse array:

Applications  (2)

Make a tridiagonal matrix:

Build a tridiagonal linear system:

Properties & Relations  (2)

Band works in SparseArray; use Normal to convert to normal lists:

The simplest case of Band is equivalent to DiagonalMatrix:

Wolfram Research (2007), Band, Wolfram Language function,


Wolfram Research (2007), Band, Wolfram Language function,


@misc{reference.wolfram_2020_band, author="Wolfram Research", title="{Band}", year="2007", howpublished="\url{}", note=[Accessed: 19-January-2021 ]}


@online{reference.wolfram_2020_band, organization={Wolfram Research}, title={Band}, year={2007}, url={}, note=[Accessed: 19-January-2021 ]}


Wolfram Language. 2007. "Band." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). Band. Wolfram Language & System Documentation Center. Retrieved from