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Concept
Conjugate transpose
Summary
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix 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
and very commonly in physics as .
For real matrices, the conjugate transpose is just the transpose, .
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
commonly used in linear algebra
(sometimes pronounced as A dagger), commonly used in quantum mechanics
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.
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 .
Official source
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