Nilpotent matrix

In mathematics, a nilpotent matrix is a square matrix that is nilpotent. For example, a matrix of the following form:

$\begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}$

This is an example of a 4×4 nilpotent matrix. Notice the non-zero superdiagonal. The Characteristic feature of this matrix is:

$N^2 = \begin{bmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ N^3 = \begin{bmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ N^4 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}$

The super-diagonal keeps 'shifting' diagonally up, until one gets the null matrix.

There is a classification theorem showing that this is typical: a nilpotent matrix is similar to a block matrix, with diagonal square blocks generalising this type, and other blocks zero.

Categories: Matrices | Linear algebra