Nilpotent matrix

From Freepedia

In mathematics, a nilpotent matrix is an n×n square matrix M such that

<math>M^q = 0\,</math>

for some positive integer q. Similarly, a nilpotent transformation is a linear transformation L with <math>L^q = 0</math> for some integer q.

These are special cases of a more general concept of nilpotence that applies not only to matrices and linear transformations but to members of rings.

Contents 1 Examples 2 Properties 3 Classification theorem 4 Flag of subspaces 5 External links

Examples

For example, a matrix of the following form:

<math>

\begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix} </math>

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

<math>

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}.

</math>

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

The corresponding nilpotent transformation L : R⁴ → R⁴ is defined by:

<math> L(x_1,x_2,x_3,x_4) = (x_2,x_3,x_4,0). \, </math>

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.

Properties

Let M be an n×n nilpotent matrix.

• The smallest integer q such that M^q = 0 is smaller than or equal to n.
• The eigenvalues of M are all zero. In fact, a matrix is nilpotent if and only if its eigenvalues are all zero.
• The characteristic polynomial of M is λⁿ.
• The determinant and trace of M are both zero.
• Every strictly upper triangular matrix or strictly lower triangular matrix is nilpotent.

Classification theorem

The above example is typical, as the following result shows. Every nilpotent matrix is similar to a block diagonal matrix

<math> \begin{bmatrix}

  N_1 & 0 & 0 & \ldots & 0 \\ 
  0 & N_2 & 0 & \ldots & 0 \\
  0 & 0 & N_3 & \ldots & 0 \\
  \vdots & \vdots & \vdots & \ddots & \vdots \\
  0 & 0 & 0 & \ldots & N_k 

\end{bmatrix} </math> where the blocks <math>N_i</math> have ones on the superdiagonal and zeros everywhere else:

<math> N_i = \begin{bmatrix}

  0 & 1 & 0 & \ldots & 0 & 0 \\
  0 & 0 & 1 & \ldots & 0 & 0 \\
  \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
  0 & 0 & 0 & \ldots & 1 & 0 \\
  0 & 0 & 0 & \ldots & 0 & 1 \\
  0 & 0 & 0 & \ldots & 0 & 0

\end{bmatrix}. </math>

This fact follows from the Jordan decomposition theorem, together with the result that every matrix similar to a nilpotent matrix is also nilpotent.

Flag of subspaces

A nilpotent transformation L on Rⁿ naturally determines a flag of subspaces

<math> \{0\} \subset \ker L \subset \ker L^2 \subset \ldots \subset \ker L^{q-1} \subset \ker L^q = U </math>

and a signature

<math> 0 = n_0 < n_1 < n_2 < \ldots < q_{k-1} < q_k = n,\qquad n_i = \dim \ker N^i. </math>

The signature characterizes L up to a linear transformation. Furthermore, it satisfies the inequalities

<math> n_{j+1} - n_j \leq n_j - n_{j-1}, \qquad \mbox{for all } j = 1,\ldots,q-1. </math>

Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.