Conjugate transpose: Difference between revisions

Revision as of 16:29, 14 September 2019

Revision as of 16:29, 14 September 2019

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==Generalizations==

The last property given above shows that if one views <math>\boldsymbol{A}</math> as a [[linear transformation]] from Euclidean [[Hilbert space]] <math> \mathbb{C}^n </math> to <math> \mathbb{C}^m ,</math> then the matrix <math>\boldsymbol{A}^\mathrm{H}</math> corresponds to the [[Hermitian adjoint|adjoint operator]] of <math>\boldsymbol A</math>. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.

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