A nilpotent operator is a linear transformation on a vector space that, when applied repeatedly, eventually results in the zero vector. In other words, there exists some positive integer n such that the nth power of the operator is the zero operator.
A linear operator is nilpotent if and only if its characteristic polynomial is x^n, where n is the dimension of the vector space on which it acts. In matrix form, a nilpotent operator will have all zeros on and above its main diagonal.
Nilpotent operators are important in the study of linear algebra and functional analysis, as they provide a way to decompose more complex operators into simpler, nilpotent ones. They also have applications in areas such as differential equations and quantum mechanics.
No, by definition, a nilpotent operator has a characteristic polynomial of x^n, which means that all of its eigenvalues are equal to 0. This also means that the only eigenvector of a nilpotent operator is the zero vector.
A nilpotent matrix is a matrix representation of a nilpotent operator. In other words, a nilpotent operator can be expressed as a nilpotent matrix when a basis for the vector space is chosen. However, not all nilpotent matrices represent nilpotent operators, as the choice of basis may affect the nilpotency of the matrix.