nilpotent matrix

The square matrixMathworldPlanetmath A is said to be nilpotent if An=AAAn times=𝟎 for some positive integer n (here 𝟎 denotes the matrix where every entry is 0).

Theorem (Characterization of nilpotent matrices).

A matrix is nilpotent iff its eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath are all 0.


Let A be a nilpotent matrixMathworldPlanetmath. Assume An=𝟎. Let λ be an eigenvalue of A. Then A𝐱=λ𝐱 for some nonzero vector 𝐱. By induction λn𝐱=An𝐱=0, so λ=0.

Conversely, suppose that all eigenvalues of A are zero. Then the chararacteristic polynomial of A: det(λI-A)=λn. It now follows from the Cayley-Hamilton theoremMathworldPlanetmath that An=𝟎. ∎

Since the determinantMathworldPlanetmath is the product of the eigenvalues it follows that a nilpotent matrix has determinant 0. Similarly, since the trace of a square matrix is the sum of the eigenvalues, it follows that it has trace 0.

One class of nilpotent matrices are the triangular matricesMathworldPlanetmath (lower or upper), this follows from the fact that the eigenvalues of a triangular matrix are the diagonal elements, and thus are all zero in the case of strictly triangular matrices.

Note for 2×2 matrices A the theorem implies that A is nilpotent iff A=𝟎 or A2=𝟎.

Also it is worth noticing that any matrix that is similar to a nilpotent matrix is nilpotent.

Title nilpotent matrix
Canonical name NilpotentMatrix
Date of creation 2013-03-22 13:05:56
Last modified on 2013-03-22 13:05:56
Owner jgade (861)
Last modified by jgade (861)
Numerical id 17
Author jgade (861)
Entry type Definition
Classification msc 15-00