Unipotent matrix

In mathematics, a unipotent matrix is a square matrix whose difference to the unit matrix is nilpotent . The unipotent matrices thus represent the unipotent elements in the ring of the square matrices .

definition

A square matrix with entries from a unitary ring is called unipotent if the matrix is nilpotent , that is, if ${\displaystyle A\in R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle A-I}$

${\displaystyle (A-I)^{m}=0}$

applies to one . Unipotent matrices are therefore the unipotent elements in the matrix ring with the zero matrix as an identity element and the identity matrix as one element . ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle R^{n\times n}}$ ${\displaystyle 0}$ ${\displaystyle I}$

Examples

A simple example of a unipotent matrix is ​​the matrix

${\displaystyle A={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$,

because it applies

${\displaystyle (A-I)^{2}={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}}$.

A more general example is formed by upper triangular matrices whose main diagonal entries are all equal to 1, i.e. matrices of the form

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

All such matrices are unipotent because it holds . Furthermore, all matrices that are similar to such a matrix are also unipotent , because it then applies ${\displaystyle (A-I)^{n}=0}$ ${\displaystyle A}$

${\displaystyle (S^{-1}AS-I)^{n}=S^{-1}(A-I)^{n}S=0}$

for any regular matrix . ${\displaystyle S\in R^{n\times n}}$

properties

Eigenvalues

A square matrix with entries from a body is unipotent if and only if its characteristic polynomial has the form ${\displaystyle A\in K^{n\times n}}$ ${\displaystyle K}$

${\displaystyle \chi _{A}(\lambda )=(\lambda -1)^{n}}$

owns. This is the case if and only if all eigenvalues ​​of the matrix are equal . ${\displaystyle 1}$

Jordan-Chevalley decomposition

Every regular matrix with entries from an algebraically closed field has a multiplicative Jordan-Chevalley decomposition of the form ${\displaystyle A}$ ${\displaystyle K}$

${\displaystyle A=D\cdot U=U\cdot D}$,

where are a diagonalizable and a unipotent matrix. Such a decomposition is clear. ${\displaystyle D}$ ${\displaystyle U}$

Potencies

The entries of the matrix powers of a real or complex unipotent matrix only grow polynomially in , da ${\displaystyle A^{k}}$ ${\displaystyle k}$

${\displaystyle A^{k}=(I+N)^{k}=I+kN+{\frac {k(k-1)}{2}}N^{2}+\ldots +{\frac {k(k-1)\cdots (k-m+2)}{(m-1)!}}N^{m-1}}$

applies, where is nilpotent with nilpotence index . Conversely, if the entries of the matrix powers of a given matrix grow at most polynomially in , the matrix is ​​unipotent. ${\displaystyle N}$ ${\displaystyle m}$ ${\displaystyle k}$

Logarithm and exponential

After the above series terminates, the matrix logarithm of a real or complex unipotent matrix exists and is itself nilpotent. The following applies to its matrix exponential

${\displaystyle e^{\log A}=A}$.

Conversely, the matrix exponential of a real or complex nilpotent matrix is unipotent and it applies accordingly ${\displaystyle N}$

${\displaystyle \log e^{N}=N}$.

literature

• Dennis S. Bernstein: Matrix Mathematics: Theory, Facts, and Formulas . Princeton University Press, 2009, ISBN 978-0-691-14039-1 .
• Ina Kersten : Linear Algebraic Groups . Universitätsverlag Göttingen, 2007, ISBN 978-3-940344-05-2 .

Individual evidence

1. ↑ Ina Kersten: Linear Algebraic Groups . Universitätsverlag Göttingen, 2007, p. 66 .
2. ^ Terence Tao : Structure and Randomness . American Mathematical Society, 2008, pp. 111 .
3. ^ ^{A b} Dennis S. Bernstein: Matrix Mathematics: Theory, Facts, and Formulas . Princeton University Press, 2009, pp. 746 .

Web links

• DA Suprunenko: Unipotent matrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Todd Rowland: Unipotent . In: MathWorld (English).