Editing Nilpotent matrix (section)

==Additional properties==
{{unordered list
| If <math>N</math> is nilpotent of index <math>k</math> , then <math>I+N</math> and <math>I-N</math> are [[invertible matrix|invertible]], where <math>I</math> is the <math>n \times n</math> [[identity matrix]]. The inverses are given by
: <math>\begin{align}
(I + N)^{-1} &= \displaystyle\sum^k_{m=0}\left(-N\right)^m = I - N + N^2 - N^3 + N^4 - N^5 + N^6 - N^7 + \cdots +(-N)^k \\
(I - N)^{-1} &= \displaystyle\sum^k_{m=0}N^m = I + N + N^2 + N^3 + N^4 + N^5 + N^6 + N^7 +  \cdots + N^k \\
\end{align}</math>

| If <math>N</math> is nilpotent, then
: <math>\det (I + N) = 1.</math>

Conversely, if <math>A</math> is a matrix and
: <math>\det (I + tA) = 1\!\,</math>
for all values of <math>t</math>, then <math>A</math> is nilpotent. In fact, since <math>p(t) = \det (I + tA) - 1</math> is a polynomial of degree <math>n</math>, it suffices to have this hold for <math>n+1</math> distinct values of <math>t</math>.
| Every [[singular matrix]] can be written as a product of nilpotent matrices.<ref>R. Sullivan, Products of nilpotent matrices, ''Linear and Multilinear Algebra'', Vol. 56, No. 3</ref>
| A nilpotent matrix is a special case of a [[convergent matrix]].
}}