Conjugate transpose

In mathematics, the conjugate transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A^* obtained from A by taking the transpose and then taking the complex conjugate of each entry. Formally

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A^*)[i,j] = \overline{A[j,i]}}

for 1 ≤ i ≤ n and 1 ≤ j ≤ m.

This definition can also be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^* \equiv {\overline A}^{T}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^T \,\!} denotes the transpose and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline A \,\!} denotes the matrix with complex conjugated entries.

Alternative names for the conjugate transpose of a matrix are adjoint matrix, Hermitian conjugate, or tranjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^* \,\!}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^H \,\!} , commonly used in linear algebra
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\dagger \,\!} , universally used in quantum field theory

In some contexts Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^* \,\!} can be used to denote the complex conjugate so care must be taken not to confuse notations.

Example

If

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\begin{bmatrix}3+i&2\\ 2-2i&i\end{bmatrix}}

then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^*=\begin{bmatrix}3-i&2+2i\\ 2&-i\end{bmatrix}.}

Basic remarks

If the entries of A are real, then A^* coincides with the transpose A^T of A. It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation.

A square matrix A is called

• Hermitian or self-adjoint if A = A^*;
• skew Hermitian if A = -A^*;
• normal if A^*A = AA^*.

Even if A is not square, the two matrices A^*A and AA^* are both Hermitian and in fact positive semi-definite.

The adjoint matrix A^* should not be confused with the adjugate adj(A) (which in older texts is also sometimes called "adjoint").

Properties of the conjugate transpose

• (A + B)^* = A^* + B^* for any two matrices A and B of the same format.
• (rA)^* = 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 (A^*) = (det A)^* and trace (A^*) = (trace A)^*
• A is invertible if and only if 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.
• <Ax,y> = <x, A^*y> for any m-by-n matrix A, any vector x in Cⁿ and any vector y in C^m. Here <.,.> denotes the ordinary Euclidean inner product (or dot product) on C^m and Cⁿ.

Generalizations

The last property given above shows that if one views A as a linear map from the Euclidean Hilbert space Cⁿ to C^m, then the matrix A^* corresponds to the adjoint operator of A. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices.

Another generalization is available: suppose A is a linear map from a complex vector space V to another W, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A. It maps the conjugate dual of W to the conjugate dual of V.

See also

• Hermitian conjugate

External links

• Conjugate transpose on Mathworld, Wolfram research
• Conjugate transpose on PlanetMath.