Conjugate transpose

# Conjugate transpose

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

A with complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

entries is the n-by-m matrix A* obtained from A by taking the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

and then taking the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

of each entry (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by
where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

. (The complex conjugate of , where a and b are reals, is .)

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, or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:
• or , commonly used in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

• (sometimes pronounced "A dagger
Dagger (typography)
A dagger, or obelisk. is a typographical symbol or glyph. The term "obelisk" derives from Greek , which means "little obelus"; from meaning "roasting spit"...

"), universally used in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

• , although this symbol is more commonly used for the Moore-Penrose pseudoinverse

In some contexts, denotes the matrix with complex conjugated entries, and thus the conjugate transpose is denoted by or .

If
then

## Basic remarks

A square matrix A with entries is called
• Hermitian or self-adjoint if A = A*, i.e.,  .
• skew Hermitian
Skew-Hermitian matrix
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to its negative. That is, the matrix A is skew-Hermitian if it satisfies the relationA^\dagger = -A,\;...

or antihermitian if A = −A*, i.e.,  .
• normal
Normal matrix
A complex square matrix A is a normal matrix ifA^*A=AA^* \ where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.If A is a real matrix, then A*=AT...

if A*A = AA*.
• unitary if A* = A-1.

Even if A is not square, the two matrices A*A and AA* are both Hermitian and in fact positive semi-definite matrices
Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....

.

## Motivation

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:

That is, denoting each complex number z by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space ) affected by complex z-multiplication on .

An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n 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 n-by-m matrix made up of complex numbers.

## Properties of the conjugate transpose

• (A + B)* = A* + B* for any two matrices A and B of the same dimensions.
• (r A)* = r*A* for any complex number r and any matrix A. Here r* refers to the complex conjugate of r.
• (AB)* = B*A* for any m-by-n matrix A and any n-by-p matrix B. Note that the order of the factors is reversed.
• (A*)* = A for any matrix A.
• If A is a square matrix, then det
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

(A*) = (det A)* and tr(A*) = (tr A)*
• A is invertible if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

A* is invertible, and in that case we have (A*)−1 = (A−1)*.
• The eigenvalues of A* are the complex conjugates of the eigenvalues of A.
• for any m-by-n matrix A, any vector x in and any vector y in . Here denotes the standard complex inner product on and .

## Generalizations

The last property given above shows that if one views A as a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

from the Euclidean Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

to , then the matrix A* corresponds to the adjoint operator