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
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 .
Examples
A simple example of a unipotent matrix is the matrix
- ,
because it applies
- .
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
- .
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
for any regular matrix .
properties
Eigenvalues
A square matrix with entries from a body is unipotent if and only if its characteristic polynomial has the form
owns. This is the case if and only if all eigenvalues of the matrix are equal .
Jordan-Chevalley decomposition
Every regular matrix with entries from an algebraically closed field has a multiplicative Jordan-Chevalley decomposition of the form
- ,
where are a diagonalizable and a unipotent matrix. Such a decomposition is clear.
Potencies
The entries of the matrix powers of a real or complex unipotent matrix only grow polynomially in , da
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.
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
- .
Conversely, the matrix exponential of a real or complex nilpotent matrix is unipotent and it applies accordingly
- .
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
- ↑ Ina Kersten: Linear Algebraic Groups . Universitätsverlag Göttingen, 2007, p. 66 .
- ^ Terence Tao : Structure and Randomness . American Mathematical Society, 2008, pp. 111 .
- ^ 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).