Self adjoint matrix

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 ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle \mathbb {K} ^{n}}$ ${\displaystyle A}$

${\displaystyle \langle Ay,x\rangle =\langle y,Ax\rangle }$

applies to all . The matrix is understood here as a linear mapping on the . ${\displaystyle x,y\in \mathbb {K} ^{n}}$ ${\displaystyle A}$ ${\displaystyle \mathbb {K} ^{n}}$

Examples

• The matrix

${\displaystyle {\begin{pmatrix}3&2+i\\2-i&1\end{pmatrix}}}$

with as the imaginary unit is self adjoint with respect to the standard scalar product due to ${\displaystyle i^{2}=-1}$ ${\displaystyle \mathbb {C} ^{n}}$

${\displaystyle {\begin{pmatrix}3&2+i\\2-i&1\\\end{pmatrix}}^{*}={\begin{pmatrix}3&{\overline {2-i}}\\{\overline {2+i}}&1\\\end{pmatrix}}={\begin{pmatrix}3&2+i\\2-i&1\\\end{pmatrix}}.}$

• The Pauli matrices

${\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

are self adjoint.

properties

A real matrix is ​​self-adjoint if and only if it is symmetric , i.e. if it holds that there ${\displaystyle A=A^{T}}$

${\displaystyle \langle Ay,x\rangle =(Ay)^{T}x=y^{T}A^{T}x=y^{T}Ax=y^{T}(Ax)=\langle y,Ax\rangle }$.

Analogous to this, a complex matrix is ​​self-adjoint if and only if it is Hermitian , i.e. if it is true that there ${\displaystyle A=A^{*}}$

${\displaystyle \langle Ay,x\rangle =(Ay)^{*}x=y^{*}A^{*}x=y^{*}Ax=y^{*}(Ax)=\langle y,Ax\rangle }$.

Any self-adjoint matrix is ​​also normal , that is, it holds

${\displaystyle A^{*}\cdot A=A\cdot A^{*}}$.

The reverse is generally not true.

See also

• Self-adjoint operator for the generalization of the term to linear operators

Individual evidence

1. ↑ Guido Walz (Ed.): Lexicon of Mathematics . in six volumes. 1st edition. tape ? . Spectrum Akademischer Verlag, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 , p. ? .

Web links

• AL Onishchik: Self-adjoint linear transformation . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).