Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
"Adjoint matrix" redirects here. For the transpose of cofactor, see
Adjugate matrix.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or ,[1] ,[2] or (often in physics) .
For real matrices, the conjugate transpose is just the transpose, .
Definition
The conjugate transpose of an matrix is formally defined by
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(Eq.1)
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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:
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:
A square matrix with entries is called
- Hermitian or self-adjoint if ; i.e., .
- Skew Hermitian or antihermitian if ; i.e., .
- Normal if .
- Unitary 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.
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.
For an explanation of the notation used here, we begin by representing complex numbers as the rotation matrix, that is,
Since
we are led to the matrix representations of the unit numbers as
A general complex number is then represented as
The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.
[3]
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.[1]
- for any matrix , i.e. Hermitian transposition is an involution.
- If is a square matrix, then where denotes the determinant of .
- If is a square matrix, then where denotes the
trace
of .
- is invertible if and only if is invertible, and in that case .
- The
eigenvalues
of are the complex conjugates of the eigenvalues
of .
- for any matrix , any vector in and any vector . Here, denotes the standard complex
inner product
on , and similarly for .
Generalizations
The last property given above shows that if one views as a
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 to another, , then the
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
References
External links