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Conjugate Transpose

The conjugate transpose of an m×n matrix A is the n×m matrix defined by

A^H=A^_^T=A^T^_.
(1)

Here, A^T is the transpose and A^_ is the conjugate matrix. Entrywise complex conjugation and transpose commute for every complex matrix.

The conjugate transpose of a matrix A is implemented in the Wolfram Language as ConjugateTranspose[A].

The conjugate transpose is also called the Hermitian conjugate or Hermitian transpose and is often called the adjoint in matrix contexts. The phrase adjoint matrix is ambiguous because it can instead mean the adjugate matrix. Several notations are in use, as summarized in the following table. The notation A^H is common in numerical linear algebra, A^* is common in operator theory, and A^| is common in physics. Since A^* can also denote the complex conjugate, the convention must be determined from context.

notationreferences
A^HThis work; Golub and van Loan (1996, p. 14), Strang (1988, p. 220)
A^*Courant and Hilbert (1989, p. 9), Lancaster and Tismenetsky (1984), Meyer (2000)
A^|Arfken (1985, p. 210), Weinberg (1995, p. xxv)

If a matrix is equal to its own conjugate transpose, it is said to be self-adjoint and is called a Hermitian.

The conjugate transpose reverses a matrix product since

(AB)^H=(AB)^T^_
(2)
=B^TA^T^_
(3)
=B^HA^H.
(4)

See also

Adjoint Matrix, Adjoint Operator, Adjugate Matrix, Complex Conjugate, Conjugate Matrix, Dagger, Hermitian Matrix, Schur Decomposition, Self-Adjoint Matrix, Transpose

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 210, 1985.Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 49, 1962.Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 14, 1996.Lancaster, P. and Tismenetsky, M. The Theory of Matrices, with Applications, 2nd ed. New York: Academic Press, 1984.Meyer, C. D. Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000.Strang, G. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988.Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 1993.Weinberg, S. The Quantum Theory of Fields, Vol. 1: Foundations. Cambridge, England: Cambridge University Press, 1995.

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Conjugate Transpose

Cite this as:

Weisstein, Eric W. "Conjugate Transpose." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConjugateTranspose.html

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