An matrix is nilpotent if for some positive integer . A nonzero nilpotent matrix must have both positive and negative entries in order for cancellation to take place in the matrix powers. The smallest for which is called the index of nilpotency. The index does not exceed , as we will see below.
Here are some examples of nilpotent matrices.
Matrix is the instance of the upper bidiagonal matrix
and . The superdiagonal of ones moves up to the right with each increase in the index of the power until it disappears off the top right corner of the matrix.
Matrix has rank and was constructed using a general formula: if with then . We simply took orthogonal vectors and .
If is nilpotent then every eigenvalue is zero, since with implies or . Consequently, the trace and determinant of a nilpotent matrix are both zero.
If is nilpotent and Hermitian or symmetric, or more generally normal (), then , since such a matrix has a spectral decomposition and the matrix is zero. It is only for nonnormal matrices that nilpotency is a nontrivial property, and the best way to understand it is with the Jordan canonical form (JCF). The JCF of a matrix with only zero eigenvalues has the form , where , where is of the form (1) and hence . It follows that the index of nilpotency is .
What is the rank of an nilpotent matrix ? The minimum possible rank is , attained for the zero matrix. The maximum possible rank is , attained when the JCF of has just one Jordan block of size . Any rank between and is possible: rank is attained when there is a Jordan block of size and all other blocks are .
Finally, while a nilpotent matrix is obviously not invertible, like every matrix it has a Moore–Penrose pseudoinverse. The pseudoinverse of a Jordan block with eigenvalue zero is just the transpose of the block: for in (1).