Nilpotent matrix

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

$M^q = 0\,$

for some positive integer q. Similarly, a nilpotent transformation is a linear transformation L with $L q = 0$ 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.

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

[edit] Examples

Consider the following matrix:

$N = \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}.$

This is an example of a 4×4 nilpotent matrix (in fact, matrices of this form are called shift matrices). 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.

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

$L(x_1,x_2,x_3,x_4) = (x_2,x_3,x_4,0). \,$

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

[edit] 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.
Over an algebraically closed field, a matrix M is nilpotent if and only if its eigenvalues are all zero. Therefore the determinant and trace of M are both zero, and nilpotent matrices are not invertible.
Suppose A and B are matrices. If A is invertible, then $A - 1 B$ is nilpotent if and only if $det(A + t B)$ does not depend on t. This follows since

$\det(A+tB)=\det A \cdot \det (I+t A^{-1}B)=\det A \cdot \prod_k (1+\lambda_k t)$

when $\lambda_1, \ldots, \lambda_n$ are eigenvalues of

A - 1 B

The characteristic polynomial of M is λⁿ.
Every strictly upper triangular matrix or strictly lower triangular matrix is nilpotent.
Every singular matrix can be written as a product of nilpotent matrices.^[1]

[edit] Classification theorem

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

$\begin{bmatrix} N_1 & 0 & 0 & \ldots & 0 \\ 0 & N_2 & 0 & \ldots & 0 \\ 0 & 0 & N_3 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & N_k \end{bmatrix}$

where the blocks $N i$ have ones on the superdiagonal and zeros everywhere else:

$N_i = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & 1 & 0 \\ 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 0 \end{bmatrix}.$