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Difference between revisions of "Self-adjoint linear transformation"

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A [[Linear transformation|linear transformation]] of a Euclidean or unitary space that coincides with its [[Adjoint linear transformation|adjoint linear transformation]]. A self-adjoint linear transformation in a Euclidean space is also called symmetric, and in a unitary space, Hermitian. A necessary and sufficient condition for the self-adjointness of a linear transformation of a finite-dimensional space is that its matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838801.png" /> in an arbitrary orthonormal basis coincides with the adjoint matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838802.png" />, that is, it is a [[Symmetric matrix|symmetric matrix]] (in the Euclidean case), or a [[Hermitian matrix|Hermitian matrix]] (in the unitary case). The eigenvalues of a self-adjoint linear transformation are real (even in the unitary case), and the eigenvectors corresponding to different eigenvalues are orthogonal. A linear transformation of a finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838803.png" /> is self-adjoint if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838804.png" /> has an orthonormal basis consisting of eigenvectors; in this basis the transformation can be described by a real diagonal matrix.
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A [[Linear transformation|linear transformation]] of a Euclidean or unitary space that coincides with its [[Adjoint linear transformation|adjoint linear transformation]]. A self-adjoint linear transformation in a Euclidean space is also called symmetric, and in a unitary space, Hermitian. A necessary and sufficient condition for the self-adjointness of a linear transformation of a finite-dimensional space is that its matrix $A$ in an arbitrary orthonormal basis coincides with the adjoint matrix $A^*$, that is, it is a [[Symmetric matrix|symmetric matrix]] (in the Euclidean case), or a [[Hermitian matrix|Hermitian matrix]] (in the unitary case). The eigenvalues of a self-adjoint linear transformation are real (even in the unitary case), and the eigenvectors corresponding to different eigenvalues are orthogonal. A linear transformation of a finite-dimensional space $L$ is self-adjoint if and only if $L$ has an orthonormal basis consisting of eigenvectors; in this basis the transformation can be described by a real diagonal matrix.
  
A self-adjoint linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838805.png" /> is non-negative (or positive semi-definite) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838806.png" /> for any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838807.png" />, and positive definite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838808.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s0838809.png" />. For a self-adjoint linear transformation in a finite-dimensional space to be non-negative (respectively, positive-definite) it is necessary and sufficient that all its eigenvalues are non-negative (respectively, positive), or that the corresponding matrix is positive semi-definite (respectively, positive-definite). In this case there is a unique non-negative self-adjoint linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s08388010.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s08388011.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s08388012.png" /> is the square root of the self-adjoint linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083880/s08388013.png" />.
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A self-adjoint linear transformation $A$ is non-negative (or positive semi-definite) if $(Ax,x)\geq0$ for any vector $x$, and positive definite if $(Ax,x)>0$ for any $x\neq0$. For a self-adjoint linear transformation in a finite-dimensional space to be non-negative (respectively, positive-definite) it is necessary and sufficient that all its eigenvalues are non-negative (respectively, positive), or that the corresponding matrix is positive semi-definite (respectively, positive-definite). In this case there is a unique non-negative self-adjoint linear transformation $B$ satisfying the condition $B^2=A$, that is, $B$ is the square root of the self-adjoint linear transformation $A$.
  
  

Latest revision as of 08:20, 3 October 2014

A linear transformation of a Euclidean or unitary space that coincides with its adjoint linear transformation. A self-adjoint linear transformation in a Euclidean space is also called symmetric, and in a unitary space, Hermitian. A necessary and sufficient condition for the self-adjointness of a linear transformation of a finite-dimensional space is that its matrix $A$ in an arbitrary orthonormal basis coincides with the adjoint matrix $A^*$, that is, it is a symmetric matrix (in the Euclidean case), or a Hermitian matrix (in the unitary case). The eigenvalues of a self-adjoint linear transformation are real (even in the unitary case), and the eigenvectors corresponding to different eigenvalues are orthogonal. A linear transformation of a finite-dimensional space $L$ is self-adjoint if and only if $L$ has an orthonormal basis consisting of eigenvectors; in this basis the transformation can be described by a real diagonal matrix.

A self-adjoint linear transformation $A$ is non-negative (or positive semi-definite) if $(Ax,x)\geq0$ for any vector $x$, and positive definite if $(Ax,x)>0$ for any $x\neq0$. For a self-adjoint linear transformation in a finite-dimensional space to be non-negative (respectively, positive-definite) it is necessary and sufficient that all its eigenvalues are non-negative (respectively, positive), or that the corresponding matrix is positive semi-definite (respectively, positive-definite). In this case there is a unique non-negative self-adjoint linear transformation $B$ satisfying the condition $B^2=A$, that is, $B$ is the square root of the self-adjoint linear transformation $A$.


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References

[a1] P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958)
How to Cite This Entry:
Self-adjoint linear transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-adjoint_linear_transformation&oldid=33469
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article