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8: Matrices dispersas

Última actualización

30 oct 2022
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- 7.3: Dimensiones más altas
- 8.1: Álgebra de matriz dispersa

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Una matriz dispersa es una matriz en la que la mayoría de las entradas son cero. Tales matrices se encuentran muy comúnmente en ecuaciones de diferencia finita. Por ejemplo, cuando discretizamos la ecuación de onda de Schrödinger 1D con condiciones de límite de Dirichlet, vimos que la matriz hamiltoniana tenía la forma tridiagonal

$\mathbf{H} = -\frac{1}{2h^2} \begin{bmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots & 1 \\ & & 1 & -2\end{bmatrix} + \begin{bmatrix}V_0 \\ & V_1 \\& & \ddots \\ & & & V_{N-1}\end{bmatrix}.$

De ahí que si hay puntos de $N$ diagonalización, la matriz hamiltoniana tiene un total de $N^{2}$ entradas, pero sólo $O(N)$ de estas entradas son diferentes de cero.

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