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Concept
Unitary Matrix
A
unitary matrix
is a
complex square matrix
whose
conjugate transpose
is also its inverse, ensuring that the matrix preserves the
inner product
in
complex vector spaces
. This property makes unitary matrices fundamental in
quantum mechanics
and various fields of
linear algebra
due to their ability to
represent rotations
and reflections without
altering vector norms
.
Concept
Unitary Transformation
A
unitary transformation
is a
linear operation
on a
complex vector space
that preserves the
inner product
, meaning it maintains the
length and angle between vectors
. This property makes
unitary transformation
s
fundamental in quantum mechanics
, where they describe the
evolution of quantum states
in a way that conserves
probability amplitudes
.
Concept
Complex Line Bundle
A
complex line bundle
is a
fiber bundle
where the fiber is a
one-dimensional complex vector space
, typically used in the context of
algebraic topology
and
differential geometry
. It provides a framework for studying
complex manifolds
and plays a crucial role in the theory of
holomorphic line bundles
and
sheaf cohomology
.
Concept
Unitary Group
The
unitary group
, denoted as U(n), is the group of
n×n unitary matrices
, which are matrices that preserve the
inner product
in
complex vector spaces
, making them central to
quantum mechanics
and various
fields of mathematics
. These matrices have the property that their inverse is equal to their
conjugate transpose
, ensuring that they maintain the norm and orthogonality of vectors upon transformation.
Concept
Hermitian Inner Product
A
Hermitian inner product
is a
complex-valued function
that generalizes the
dot product
to
complex vector spaces
, ensuring that the
inner product of a vector
with itself is always
real and non-negative
. It is a crucial tool in
quantum mechanics
and
linear algebra
, providing the framework for defining orthogonality, norms, and angles in
complex spaces
.
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