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Nilpotent Matrix Checker

Check if matrix is nilpotent (A^k = 0)

Enter 2x2 Matrix

Nilpotent Matrix Definition

A is nilpotent ⟺ A^k = 0 for some k ∈ ℕ

Index of nilpotency = smallest such k

For n×n: index ≤ n

All eigenvalues = 0 (characteristic = λⁿ)

A nilpotent matrix becomes the zero matrix when raised to some positive power. The index of nilpotency is the smallest such power. Nilpotent matrices have only zero as their eigenvalue.

⚠ All eigenvalues of a nilpotent matrix are zero. This is necessary but not sufficient for nilpotency.

What is a Nilpotent Matrix?

A nilpotent matrix is a square matrix A for which some power A^k equals the zero matrix for some positive integer k. The smallest such k is called the index of nilpotency. Strictly upper (or lower) triangular matrices with zeros on the diagonal are nilpotent.

Zero Eigenvalues

All eigenvalues are 0. Characteristic polynomial = λⁿ.

Trace = 0

Since all eigenvalues are 0, trace is always 0.

Index ≤ n

For n×n matrix, nilpotency index is at most n.

Jordan Form

Example: N = [[0,1],[0,0]] (strictly upper triangular)
1. N² = [[0,0],[0,0]] = 0 ✓
2. N is nilpotent with index 2 ✓
3. N projects vectors onto y-axis, then kills them.

Applications

Lie Algebras Jordan Decomposition Differential Equations Physics Linear Algebra

Frequently Asked Questions

What is a nilpotent matrix?▼

Matrix where A^k = 0 (zero matrix) for some positive integer k. Index is smallest such k. All eigenvalues are 0.

Examples of nilpotent matrices?▼

Strictly upper triangular matrices. [[0,1],[0,0]] is nilpotent with index 2.

Properties of nilpotent matrices?▼

Trace = 0, determinant = 0, all eigenvalues = 0, index ≤ n for n×n matrix.

How to check nilpotency?▼

Compute powers A², A³, ... until reaching zero matrix or exceeding size n.

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