Bandwidth of a matrix

Definition

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

The left half-bandwidth of $A < m a t h > i s d e f i n e d a s t h e s m a l l e s t p o s i t i v e i n t e g e r < m a t h > k_{1}$ such that $a_{i j} = 0$ 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 $a_{i j} = 0$ 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	$n$