Nilpotent matrix

The nilpotent matrix and the nilpotent endomorphism are concepts from the mathematical subfield of linear algebra.

Definition

A square matrix is called nilpotent when their powers provides the zero matrix:

Correspondingly, one refers to a vector space endomorphism as nilpotent if there exists a number such that the zero mapping is. The smallest natural number which fulfills this criterion are known as Nilpotenzgrad or Nilpotenzindex. Between nilpotent matrices and nilpotent endomorphisms there is the following relation: for every nilpotent matrix exactly one nilpotent endomorphism with the property that the representation of matrix is. Are available for each nilpotent endomorphism exactly a nilpotent matrix representation.

Equivalent definitions

For a square matrix with rows and columns following statements are equivalent:

• Is nilpotent.
• There is a with and. Then is nilpotent with the Nilpotenzgrad.
• The characteristic polynomial of has the form.
• The minimal polynomial has the form of a.
• For an invertible matrix.

Example

An example of a matrix with nilpotent Nilpotenzgrad 2 is the matrix

There.

Properties of nilpotent matrices

If a matrix is nilpotent with Nilpotenzgrad k, then ...

• It only has the eigenvalue zero. This follows directly from the form of the characteristic polynomial whose zeros are the eigenvalues ​​.
• Is not invertible it because all their eigenvalues ​​are zero.
• Is either or she is not diagonalizable, since all diagonal matrices are not equal nilpotent.
• The determinant is zero.
• Is the trace zero.
• She has not full rank, that is, its column vectors are linearly dependent. There are not nilpotent all square matrices simultaneously with linearly dependent columns.
• Is invertible (the identity matrix ), because it is.

Since a nilpotent matrix is ​​a special case of a nilpotent element of a ring, in the article " nilpotent element " made ​​general statements also apply here.