Matriz nilpotente

En álgebra lineal, una matriz se dice que es nilpotente si existe tal que . Se llama índice de nilpotencia o se dice que es de índice (o de orden) y se define como .

Teorema

Si es una matriz nilpotente, entonces su determinante es cero. Que el determinante sea cero es una condición necesaria para ser una matriz nilpotente, aunque no es una condición suficiente.

Demostración

Si A es una matriz nilpotente de orden k, se sigue que . Por lo tanto, . Luego, por lo que .

El recíproco no es cierto; por ejemplo, la matriz

tiene determinante igual a cero, pero no es nilpotente. Una condición necesaria y suficiente es que la matriz no tenga autovalores diferentes de cero, en ese caso la matriz es nilpotente.

Ejemplos

La matriz

es nilpotente, ya que M2 = 0. En términos más generales, cualquier matriz triangular con ceros a lo largo de la diagonal principal es nilpotente. Por ejemplo, la matriz

es nilpotente, con

Aunque los ejemplos anteriores tienen un gran número de ceros en las entradas, no todas las matrices nilpotentes lo tienen. Por ejemplo, las matrices

ambas elevadas al cuadrado son cero, aunque ninguna matriz tiene ceros en las entradas.

Propiedades adicionales

  • Si N es nilpotente, entonces I + N es invertible, donde I es la matriz identidad de orden n (n × n). El inverso viene dado por:
donde sólo un número finito de términos del desarrollo anterior es diferente de cero.
  • Si N es nilpotente, entonces
donde I es de nuevo la matriz identidad de orden n. Recíprocamente, si A es una matriz y
para todos los valores de t, entonces A es nilpotente.

Generalizaciones

Un operador lineal T es localmente nilpotente si para todo vector v, existe un k tal que

Para operadores sobre espacios vectoriales de dimensión finita, la nilpotencia local equivale a la nilpotencia convencional.

Referencias

  1. R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

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