PlanetMath: nilpotent matrix

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nilpotent matrix

(Definition)

The square matrix is said to be nilpotent if $A^n = \underbrace{AA\cdots A}_{\textrm{n times}} = \mathbf{0}$ for some positive integer (here $\mathbf{0}$ denotes the matrix where every entry is 0).

Theorem 1 (Characterization of nilpotent matrices) A matrix is nilpotent iff its eigenvalues are all 0.

Proof. Let

be a nilpotent matrix. Assume $A^n = \mathbf{0}$ . Let $\lambda$ be an eigenvalue of

. Then $A\mathbf{x} = \lambda \mathbf{x}$ for some nonzero vector $\mathbf{x}$ . By induction $\lambda^n \mathbf{x} = A^n \mathbf{x} = 0$ , so $\lambda = 0$ .

Conversely, suppose that all eigenvalues of are zero. Then the chararacteristic polynomial of : $\det(\lambda I - A) = \lambda^n$ . It now follows from the Cayley-Hamilton theorem that $A^n = \mathbf{0}$ . $\qedsymbol$

Since the determinant 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 strictly triangular matrices (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 $2\times2$ matrices the theorem implies that is nilpotent iff $A=\mathbf{0}$ or $A^2=\mathbf{0}$ .

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

"nilpotent matrix" is owned by jgade.

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Cross-references: similar, implies, strictly, diagonal, triangular matrix, class, sum, trace, product, determinant, Cayley-Hamilton theorem, polynomial, induction, vector, eigenvalue, eigenvalues, iff, matrix, integer, positive, nilpotent, square matrix
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This is version 14 of nilpotent matrix, born on 2002-10-16, modified 2006-06-13.
Object id is 3520, canonical name is NilpotentMatrix.
Accessed 19635 times total.

Classification:

AMS MSC:

15-00 (Linear and multilinear algebra; matrix theory :: General reference works )

Pending Errata and Addenda

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