# Nilpotent matrix

In linear algebra, a nilpotent matrix is a square matrix N such that

N

k

=
0

{\displaystyle N^{k}=0\,}

for some positive integer k. The smallest such k is sometimes called the degree or index of N.[1]
More generally, a nilpotent transformation is a linear transformation L of a vector space such that Lk = 0 for some positive integer k (and thus, Lj = 0 for all j ≥ k).[2][3][4] Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Contents

1 Examples
2 Characterization
3 Classification
4 Flag of subspaces
6 Generalizations
7 Notes
8 References

Examples
The matrix

M
=

[

0

1

0

0

]

{\displaystyle M={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}

is nilpotent, since M2 = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent, with degree

n

{\displaystyle \leq n}

. For example, the matrix

N
=

[

0

2

1

6

0

0

1

2

0

0

0

3

0

0

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