In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an element ''x'' of a
*-algebra is normal if it satisfies
This definition stems from the definition of a normal linear operator in
functional analysis, where a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
''A'' from a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
into itself is called unitary if
where the
adjoint of ''A'' is ''A'' and the domain of ''A'' is the same as that of ''A''. See
normal operator for a detailed discussion. If the Hilbert space is finite-dimensional and an
orthonormal basis has been chosen, then the operator ''A'' is normal if and only if the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
describing ''A'' with respect to this basis is a
normal matrix.
See also
*
*
*
References
*
*
*
Abstract algebra
Linear algebra
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