8.4 Special Sparsity Notation

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Properties: 8.3 Sparse Matrices
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1 - ⋆ - \times - 0

Notation

Many features of matrix algebra rely on exploiting special structure in the location of nonzero elements. We will identify the sparsity structure of a matrix by specifying the location of zero & nonzero entries of our matrix.

$1$ means the entry here is equal to $1$
$⋆$ means the entry must be nonzero
$\times$ means the entry is any real number (nonzero or zero)
$0$ means the entry must be 0

Below are some examples of general patterns we use to identify the sparsity structure of matrices.

[\begin{matrix} 1 & 0 & 0 & 0 \\ \times & 1 & 0 & 0 \\ \times & \times & 1 & 0 \\ \times & \times & \times & 1 \end{matrix}], [\begin{matrix} ⋆ & \times & \times & \times & \times \\ 0 & ⋆ & \times & \times & \times \\ 0 & 0 & ⋆ & \times & \times \\ 0 & 0 & 0 & ⋆ & \times \\ 0 & 0 & 0 & 0 & ⋆ \end{matrix}], [\begin{matrix} \times & \times & \times & \times & \times \\ ⋆ & \times & \times & \times & \times \\ 0 & ⋆ & \times & \times & \times \\ 0 & 0 & ⋆ & \times & \times \\ 0 & 0 & 0 & ⋆ & \times \\ 0 & 0 & 0 & 0 & ⋆ \end{matrix}], [\begin{matrix} \times & \times & 0 & 0 \\ \times & \times & \times & 0 \\ 0 & \times & \times & \times \\ 0 & 0 & \times & \times \end{matrix}]