Linear and multilinear algebra; matrix theory Definition

Nilpotent matrix

Definition

Let $R$ be a ring and let $N$ be an $n \times n$ square matrix over $R$ . The matrix $N$ is nilpotent when $N^k = 0$ for some integer $k \ge 1$ . The least such $k$ is the index of $N$ .

Remarks

The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent of index $2$ . Each strictly upper triangular matrix is nilpotent. Over a field, an $n \times n$ matrix $N$ is nilpotent if and only if its characteristic polynomial is $x^n$ ; then $N^n = 0$ . The only eigenvalue of a nilpotent matrix over a field is $0$ , so its trace and determinant are $0$ .

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Matrix definition
Square matrix definition
Nilpotent matrix definition

Tag	`0419GJC84P`
URI	`m2://0419GJC84P`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Nilpotent matrix
Uses	Square matrix

Nilpotent matrix

Definition

Remarks

Reading path

Further reading

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