1.7: Matrices de permutación
- Page ID
- 119130
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Una matriz de permutación es otro tipo de matriz ortogonal. Cuando se multiplica a la izquierda, una matriz de\(n\) permutación\(n\) -by- reordena las filas de una\(n\) matriz\(n\) -by-, y cuando se multiplica a la derecha, reordena las columnas. Por ejemplo, deje que la cadena\(12\) represente el orden de las filas (columnas) de una matriz de dos por dos. Entonces las permutaciones de las filas (columnas) vienen dadas por\(12\) y\(21\). La primera permutación no es ninguna permutación en absoluto, y la matriz de permutación correspondiente es simplemente la matriz de identidad. La segunda permutación de las filas (columnas) se logra mediante
\[\left(\begin{array}{cc}0&1\\1&0\end{array}\right)\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=\left(\begin{array}{cc}c&d\\a&b\end{array}\right),\quad\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\left(\begin{array}{cc}0&1\\1&0\end{array}\right)=\left(\begin{array}{cc}b&a\\d&c\end{array}\right).\nonumber \]
Las filas (columnas) de una\(3\) matriz\(3\) -by- tiene\(3! = 6\) posibles permutaciones, a saber\(123, 132, 213, 231, 312, 321\). Por ejemplo, la permutación de fila\(312\) se obtiene mediante
\[\left(\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{array}\right)\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)=\left(\begin{array}{ccc}g&h&i\\a&b&c\\d&e&f\end{array}\right).\nonumber \]
Evidentemente, la matriz de permutación se obtiene permutando las filas correspondientes de la matriz de identidad. Debido a que las columnas y filas de la matriz de identidad son ortonormales, la matriz de permutación es una matriz ortogonal.