Theorem (Characterization of nilpotent matrices).
A matrix is nilpotent iff its eigenvalues are all 0.
Let be a nilpotent matrix. Assume . Let be an eigenvalue of . Then for some nonzero vector . By induction , so .
Since the determinant is the product of the eigenvalues it follows that a nilpotent matrix has determinant 0. Similarly, since the trace of a square matrix is the sum of the eigenvalues, it follows that it has trace 0.
One class of nilpotent matrices are the http://planetmath.org/node/4381strictly triangular matrices (lower or upper), this follows from the fact that the eigenvalues of a triangular matrix are the diagonal elements, and thus are all zero in the case of strictly triangular matrices.
Note for matrices the theorem implies that is nilpotent iff or .
Also it is worth noticing that any matrix that is similar to a nilpotent matrix is nilpotent.
|Date of creation||2013-03-22 13:05:56|
|Last modified on||2013-03-22 13:05:56|
|Last modified by||jgade (861)|