Lumped element model: Difference between revisions
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In [[mathematics]], an element ''x'' of a [[star-algebra]] is '''self-adjoint''' if <math>x^*=x</math>. | |||
A collection ''C'' of elements of a star-algebra is '''self-adjoint''' if it is closed under the [[Involution (mathematics)|involution]] operation. For example, if <math>x^*=y</math> then since <math>y^*=x^{**}=x</math> in a star-algebra, the set {''x'',''y''} is a self-adjoint set even though ''x'' and ''y'' need not be self-adjoint elements. | |||
In [[functional analysis]], a [[linear operator]] ''A'' on a [[Hilbert space]] is called '''self-adjoint''' if it is equal to its own [[adjoint operator|adjoint]] ''A''* and that the domain of ''A'' is the same as that of ''A''*. See [[self-adjoint operator]] for a detailed discussion. If the Hilbert space is finite-dimensional and an [[orthonormal basis]] has been chosen, then the operator ''A'' is self-adjoint if and only if the [[matrix (mathematics)|matrix]] describing ''A'' with respect to this basis is [[Hermitian matrix|Hermitian]], i.e. if it is equal to its own [[conjugate transpose]]. Hermitian matrices are also called '''self-adjoint'''. | |||
==See also== | |||
*[[Symmetric matrix]] | |||
*[[Hermitian]] | |||
==References== | |||
*{{cite book |authorlink=Michael C. Reed |first=M. |last=Reed |authorlink2=Barry Simon |first2=B. |last2=Simon |title=Methods of Mathematical Physics |others=Vol 2 |publisher=Academic Press |year=1972 |isbn= }} | |||
*{{cite book |authorlink=Gerald Teschl |first=G. |last=Teschl |title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators |publisher=American Mathematical Society |location=Providence |year=2009 |url=http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ }} | |||
{{DEFAULTSORT:Self-Adjoint}} | |||
[[Category:Abstract algebra]] | |||
[[Category:Linear algebra]] | |||
Revision as of 11:14, 19 January 2014
In mathematics, an element x of a star-algebra is self-adjoint if .
A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements.
In functional analysis, a linear operator A on a Hilbert space is called self-adjoint if it is equal to its own adjoint A* and that the domain of A is the same as that of A*. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint.
See also
References
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