Conjugate Matrix

A conjugate matrix is a matrix A^_ obtained from a given matrix A by taking the complex conjugate of each element of A (Courant and Hilbert 1989, p. 9), i.e.,


The notation A^* is sometimes also used, which can lead to confusion since this symbol is also used to denote the conjugate transpose.

Using a matrix X in a similarity transformation X^(-1)AX of a given matrix A is also known as conjugating A by X. In this case, B=X^(-1)AX and A are known as similar matrices.

See also

Complex Conjugate, Conjugate Transpose, Similar Matrices, Similarity Transformation

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Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 355-356, 1985.Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, pp. 12-13, 1962.Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989.

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Conjugate Matrix

Cite this as:

Weisstein, Eric W. "Conjugate Matrix." From MathWorld--A Wolfram Web Resource.

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