Nilpotent Matrix - Characterization

Characterization

For an n × n square matrix N with real (or complex) entries, the following are equivalent:

1. N is nilpotent.
2. The minimal polynomial for N is λk for some positive integer k ≤ n.
3. The characteristic polynomial for N is λn.
4. The only (complex) eigenvalue for N is 0.
5. tr(Nk) = 0 for all k > 0.

The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton's identities)

This theorem has several consequences, including:

• The degree of an n × n nilpotent matrix is always less than or equal to n. For example, every 2 × 2 nilpotent matrix squares to zero.
• The determinant and trace of a nilpotent matrix are always zero.
• The only nilpotent diagonalizable matrix is the zero matrix.