# maximal width of nne of matrix-rows

The smallest $b\in\mathbb{Z}_0$ such that $M_{ij} = 0$ for $|i-j| > b$ is the bandwidth of a matrix $M$.

Is there also a standard name for the smallest $w\in\mathbb{Z}_{\geq 0}$ such that $M_{ki}\neq0$ and $M_{kj}\neq0$ implies $|i-j| < w$ for every row $k$ of the matrix $M\in\mathbb{R}^{m\times n}$, i.e., \begin{align*} \min\left\{w\in\mathbb{Z} \mid w\geq 0 \wedge \forall k\in\mathbb{N}_{m} \forall i,j\in\mathbb{N}_n : (M_{ki}\neq 0\wedge M_{kj}\neq 0)\Rightarrow |i-j|<w \right\}. \end{align*} In other words $w$ is the width of the largest row-section of a matrix starting and ending with a nonzero element.

Maybe I would say pattern width of the matrix if I had to choose a name. But I would rather use something standard.

I need this for description of an efficient least-squares-algorithm exploiting Givens rotations where the order of the equations and therefore the bandwidth is rather irrelevant but that constant $w$ mentioned above is crucial for the performance.

Let $l_i$ and $u_i$ denote the column numbers of the first and last non-zero element in row $i$ of a given $p\times q$ matrix $A$ $(p\geq q)$; then the bandwidth BAND of this matrix is defined as $\max\left(u_i-l_i+1\mid 1\leq i\leq p\right)$.