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 ...
, the conjugate transpose, also known as the Hermitian transpose, of an
complex 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 ...
is an
matrix obtained by
transposing and applying
complex conjugate on each entry (the complex conjugate of
being
, for real numbers
and
). It is often denoted as
or
or
.
[
H. W. Turnbull, A. C. Aitken,
"An Introduction to the Theory of Canonical Matrices,"
1932.
]
For
real matrices, the conjugate transpose is just the transpose,
.
Definition
The conjugate transpose of an
matrix
is formally defined by
where the subscript
denotes the
-th entry, for
and
, and the overbar denotes a scalar complex conjugate.
This definition can also be written as
:
where
denotes the transpose and
denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix
can be denoted by any of these symbols:
*
, commonly used in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
*
, commonly used in linear algebra
*
(sometimes pronounced as ''A
dagger
A dagger is a fighting knife with a very sharp point and usually two sharp edges, typically designed or capable of being used as a thrusting or stabbing weapon.State v. Martin, 633 S.W.2d 80 (Mo. 1982): This is the dictionary or popular-use def ...
''), commonly used in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
*
, although this symbol is more commonly used for the
Moore–Penrose pseudoinverse
In some contexts,
denotes the matrix with only complex conjugated entries and no transposition.
Example
Suppose we want to calculate the conjugate transpose of the following matrix
.
:
We first transpose the matrix:
:
Then we conjugate every entry of the matrix:
:
Basic remarks
A square matrix
with entries
is called
*
Hermitian or
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a st ...
if
; i.e.,
.
*
Skew Hermitian or antihermitian if
; i.e.,
.
*
Normal if
.
*
Unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
if
, equivalently
, equivalently
.
Even if
is not square, the two matrices
and
are both Hermitian and in fact
positive semi-definite matrices.
The conjugate transpose "adjoint" matrix
should not be confused with the
adjugate,
, which is also sometimes called ''adjoint''.
The conjugate transpose of a matrix
with
real entries reduces to the
transpose of
, as the conjugate of a real number is the number itself.
Motivation
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by
real matrices, obeying matrix addition and multiplication:
:
That is, denoting each ''complex'' number
by the ''real''
matrix of the linear transformation on the
Argand diagram (viewed as the ''real'' vector space
), affected by complex ''
''-multiplication on
.
Thus, an
matrix of complex numbers could be well represented by a
matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an
matrix made up of complex numbers.
Properties of the conjugate transpose
*
for any two matrices
and
of the same dimensions.
*
for any complex number
and any
matrix
.
*
for any
matrix
and any
matrix
. Note that the order of the factors is reversed.
*
for any
matrix
, i.e. Hermitian transposition is an
involution.
* If
is a square matrix, then
where
denotes the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of
.
* If
is a square matrix, then
where
denotes the
trace of
.
*
is
invertible if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
is invertible, and in that case
.
* The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of
are the complex conjugates of the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of
.
*
for any
matrix
, any vector in
and any vector
. Here,
denotes the standard complex
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on
, and similarly for
.
Generalizations
The last property given above shows that if one views
as a
linear transformation from
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 ...
to
then the matrix
corresponds to the
adjoint operator of
. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.
Another generalization is available: suppose
is a linear map from a complex
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
to another,
, then the
complex conjugate linear map
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other ...
as well as the
transposed linear map are defined, and we may thus take the conjugate transpose of
to be the complex conjugate of the transpose of
. It maps the conjugate
dual of
to the conjugate dual of
.
See also
*
Complex dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebr ...
*
Hermitian adjoint
*
Adjugate matrix
References
External links
* {{springer, title=Adjoint matrix, id=p/a010850
Linear algebra
Matrices