Bandwidth of a matrix

Definition

Suppose $n$ is a positive integer and $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is a $n\times n$ square matrix.

The left half-bandwidth of $A<math>isdefinedasthesmallestpositiveinteger<math>k_{1}$ such that $a_{ij}=0$ whenever $j<i-k_{1}$. In other words, entries that are more than $k_{1}$ positions below the main diagonal are zero.

The right half-bandwidth of $A$ is defined as the smallest positive integer $k_{2}$ such that $a_{ij}=0$ whenever $j<i+k_{2}$. In other words, entries that are more than $k_{2}$ positions above the main diagonal are zero.

The banwidth is defined as $k_{1}+k_{2}+1$, where $k_{1},k_{2}$ are the left and right half-bandwidths respectively.

Ambiguity with terminology

When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is at most that quantity, not that it is necessarily exactly equal to that quantity.

Particular cases

Matrix type (all matrices are $n\times n$)	Left half-bandwidth (upper bound)	Right half-bandwidth (upper bound)	Bandwidth (upper bound)
diagonal matrix	0	0	1
tridiagonal matrix	1	1	3
pentadiagonal matrix	2	2	5
upper triangular matrix	0	$n-1$	$n$
lower triangular matrix	$n-1$	0	$n$