Nile potent matrix

The nilpotent matrix and the nilpotent endomorphism are terms from the mathematical branch of linear algebra .

definition

A square matrix is called nilpotent if one of its powers results in the zero matrix :

${\displaystyle A^{k}=0}$ for a ${\displaystyle k\in \mathbb {N} }$

Similarly, a vector space endomorphism is said to be nilpotent if there is a number such that it is the null map . The smallest natural number that fulfills this criterion is called the Nilpotency degree or Nilpotenzindex . There is the following relationship between nilpotent matrices and nilpotent endomorphisms: For every nilpotent matrix , the left-hand multiplication of this matrix by column vectors is a nilpotent endomorphism. Conversely, every representation matrix of a nilpotent endomorphism is nilpotent. ${\displaystyle f}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle f\,^{k}}$ ${\displaystyle k}$ ${\displaystyle A}$

Equivalent Definitions

For a square matrix with rows and columns, the following statements are equivalent: ${\displaystyle A}$ ${\displaystyle n}$

• ${\displaystyle A}$ is nilpotent.
• There is a with and . Then is nilpotent with the degree of nilpotency . ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle A^{k}=0}$ ${\displaystyle A^{k-1}\neq 0}$ ${\displaystyle A}$ ${\displaystyle k}$
• The characteristic polynomial of has the form . ${\displaystyle A}$ ${\displaystyle \chi _{A}(\lambda )=\det(\lambda I-A)=\lambda ^{n}}$
• The minimal polynomial of has the form for a . ${\displaystyle A}$ ${\displaystyle m_{A}(\lambda )=\lambda ^{k}}$ ${\displaystyle k>0}$
• ${\displaystyle A}$is similar to a strict triangular matrix , that is, there is an invertible matrix such that: ${\displaystyle P}$

${\displaystyle \,\,\,\,\,\,\,A=P^{-1}{\begin{pmatrix}0&b_{1,2}&\cdots &b_{1,n}\\0&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &b_{n-1,n}\\0&\cdots &0&0\end{pmatrix}}P}$

• Especially for matrices over or other algebraically closed fields it is true that they are nilpotent if and only if their only eigenvalue is 0. ${\displaystyle \mathbb {C} }$

example

An example of a nilpotent matrix with nilpotency degree 2 is the matrix

${\displaystyle A={\begin{pmatrix}5&-3&2\\15&-9&6\\10&-6&4\end{pmatrix}}}$

there . ${\displaystyle A^{2}=0}$

Properties of nilpotent matrices

If a matrix is nilpotent with degree k, then ... ${\displaystyle A}$

• it only has the eigenvalue zero. This follows directly from the form of the characteristic polynomial , the zeros of which are the eigenvalues. ${\displaystyle \chi _{A}(\lambda )=\lambda ^{n}}$
• it is not invertible , since it has the eigenvalue zero and therefore its core is not trivial.
• is either or it is not diagonalizable , since all diagonal matrices are unequal and not nilpotent. ${\displaystyle A=0}$ ${\displaystyle 0}$
• is the determinant zero: . ${\displaystyle \det(A)=0}$
• the track is zero.
• does it not have a full rank, d. H. their column vectors are linearly dependent . However, not all square matrices with linearly dependent columns are also nilpotent at the same time.
• is invertible ( is the identity matrix ) because it is . ${\displaystyle I-A}$ ${\displaystyle I}$ ${\displaystyle (I-A)\left(I+A+A^{2}+...+A^{k-1}\right)=I-A^{k}=I}$

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

Jordan-Chevalley decomposition

Every endomorphism of a finite-dimensional vector space over an algebraically closed body can be clearly written as the sum of a diagonalizable and a nilpotent endomorphism. This decomposition is called the Jordan-Chevalley decomposition and is essentially a consequence of the existence of the Jordan normal form .

literature

• Gerd Fischer : Linear Algebra. (An introduction for new students) (=  Vieweg Studium. Basic course in mathematics ). 14th, revised edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 , pp. 384 .