In
mathematics, an element ''x'' of a
*-algebra is unitary if it satisfies
In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, a
linear operator ''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 natu ...
into itself is called unitary if it is invertible and its inverse is equal to its own
adjoint ''A'' and that the domain of ''A'' is the same as that of ''A''. See
unitary operator for a detailed discussion. If the Hilbert space is finite-dimensional and an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...
has been chosen, then the operator ''A'' is unitary 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'' (franchi ...
describing ''A'' with respect to this basis is a
unitary matrix
In linear algebra, a Complex number, complex Matrix (mathematics), square matrix is unitary if its conjugate transpose is also its Invertible matrix, inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, esp ...
.
See also
*
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*
References
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{{DEFAULTSORT:Self-Adjoint
Abstract algebra
Linear algebra