Nilpotent Matrix — Definition, Formula & Examples

A nilpotent matrix is a square matrix

A

such that

A^k = 0

for some positive integer

k

. In other words, if you keep multiplying the matrix by itself, you eventually get the zero matrix.

A square matrix $A \in M_n(\mathbb{F})$ is nilpotent if there exists a positive integer $k$ such that $A^k = \mathbf{0}$ , where $\mathbf{0}$ is the $n \times n$ zero matrix. The smallest such $k$ is called the index of nilpotency (or nilpotency index) of $A$ , and it satisfies $k \leq n$ .

Key Formula

A^k = \mathbf{0}

Where:

$A$ = An $n \times n$ square matrix
$k$ = The smallest positive integer for which $A^k$ equals the zero matrix (the nilpotency index)
$\mathbf{0}$ = The $n \times n$ zero matrix

How It Works

To check whether a matrix is nilpotent, compute successive powers

A^2, A^3, \ldots

and see if any equals the zero matrix. For an

n \times n

nilpotent matrix, the nilpotency index is at most

n

, so you never need to check beyond

A^n

. Every nilpotent matrix has all eigenvalues equal to zero, which means its determinant and trace are both zero. Conversely, if a matrix has all zero eigenvalues, it is nilpotent.

Worked Example

Problem: Determine whether the matrix

A = \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}

is nilpotent, and if so, find its nilpotency index.

Step 1: Compute

A^2

by multiplying

A

by itself.

A^2 = \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

Step 2: Since

A^2 \neq \mathbf{0}

, compute

A^3 = A^2 \cdot A

A^3 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

Answer:

A

is nilpotent with nilpotency index

k = 3

, since

A^3 = \mathbf{0}

and

A^2 \neq \mathbf{0}

Why It Matters

Nilpotent matrices appear naturally in Jordan normal form, where every Jordan block with eigenvalue zero is nilpotent. They also arise in solving systems of linear differential equations and in computing matrix exponentials, since the exponential of a nilpotent matrix is a finite polynomial rather than an infinite series.

Common Mistakes

Mistake: Assuming that if

A^2 \neq \mathbf{0}

, then

A

is not nilpotent.

Correction: The nilpotency index can be any integer from 1 up to

n

. You must check all powers up to

A^n

before concluding that

A

is not nilpotent.