Nilpotent matrices admit two equivalent characterizations.
1. A square matrix whose eigenvalues are all 0.
2. A square matrix 
such that 
is the zero matrix 
for some positive integer matrix
power 
,
known as the index (Ayres 1962, p. 11).
Nilpotent Matrix
See also
Eigenvalue, Idempotent Matrix, Matrix Polynomial, Square MatrixExplore with Wolfram|Alpha
References
Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 11, 1962.Referenced on Wolfram|Alpha
Nilpotent MatrixCite this as:
Weisstein, Eric W. "Nilpotent Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NilpotentMatrix.html