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# Definition of band matrix

Put content here**Definition:** A **band matrix** is a sparse matrix whose nonzero entries are confined to a diagonal band around the main diagonal. A matrix with **lower bandwidth** $p$ and **upper bandwidth** $q$ satisfies:
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$$a_{ij} = 0 \quad \text{whenever} \quad i - j > p \; \text{or} \; j - i > q$$
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The **total bandwidth** is $p + q + 1$.
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**Special cases:**
- Tridiagonal ($p = q = 1$): nonzero only on main diagonal and adjacent diagonals
- Diagonal ($p = q = 0$): nonzero only on main diagonal
- Upper bidiagonal ($p = 0, q = 1$)
- Lower bidiagonal ($p = 1, q = 0$)
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**Example (tridiagonal):**
$$A = \begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix}$$
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Band matrices arise frequently in numerical solutions of differential equations. Specialized algorithms can solve band systems in $O(n)$ time vs $O(n^3)$ for general matrices.

# Parents

* Particular types of matrices