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Self-adjoint operator
In
mathematics
, a
self-adjoint operator
on a
complex vector space
V
with
inner product
is a
linear map
A
(from
V
to itself) that is its own
adjoint
: . If
V
is finite-dimensional with a given
orthonormal basis
, this is equivalent to the condition that the
matrix
of
A
is
Hermitian
, i.e., equal to its
conjugate transpose
A
*. By the finite-dimensional
spectral theorem
,
V
has an
orthonormal basis
such that the matrix of
A
relative to this basis is a
diagonal matrix
with entries in the
real numbers
. In this article, we consider
generalizations
of this
concept
to operators on
Hilbert spaces
of arbitrary dimension.
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