Editing Nilpotent matrix (section)

==Characterization==
{{Unreferenced section|date=May 2018}}
For an <math>n \times n</math> square matrix <math>N</math> with [[real number|real]] (or [[complex number|complex]]) entries, the following are equivalent:
* <math>N</math> is nilpotent.
* The [[characteristic polynomial]] for <math>N</math> is <math>\det \left(xI - N\right) = x^n</math>.
* The [[minimal polynomial (linear algebra)|minimal polynomial]] for <math>N</math> is <math>x^k</math> for some positive integer <math>k \leq n</math>.
* The only complex eigenvalue for <math>N</math> is 0.
The last theorem holds true for matrices over any [[field (mathematics)|field]] of characteristic 0 or sufficiently large characteristic. (cf. [[Newton's identities]])

This theorem has several consequences, including:
* The index of an <math>n \times n</math> nilpotent matrix is always less than or equal to <math>n</math>.  For example, every <math>2 \times 2</math> nilpotent matrix squares to zero.
* The [[determinant]] and [[trace (linear algebra)|trace]] of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be [[invertible matrix|invertible]].
* The only nilpotent [[diagonalizable matrix]] is the zero matrix.

See also: [[Jordan–Chevalley decomposition#Nilpotency criterion]].