A self-adjoint matrix is an object from the mathematical branch of linear algebra . It is a special type of square matrix . If the coefficients of a self-adjoint matrix are real , then it is a symmetric matrix and if the coefficients are complex , then it is a Hermitian matrix .
definition
Be the real or complex number field and be the standard scalar product on . A matrix is called self-adjoint if
applies to all . The matrix is understood here as a linear mapping on the .
Examples
- with as the imaginary unit is self adjoint with respect to the standard scalar product due to
- are self adjoint.
properties
A real matrix is self-adjoint if and only if it is symmetric , i.e. if it holds that there
-
.
Analogous to this, a complex matrix is self-adjoint if and only if it is Hermitian , i.e. if it is true that there
-
.
Any self-adjoint matrix is also normal , that is, it holds
-
.
The reverse is generally not true.
See also
Individual evidence
-
↑ Guido Walz (Ed.): Lexicon of Mathematics . in six volumes. 1st edition. tape ? . Spectrum Akademischer Verlag, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 , p. ? .
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