Conjugate Transpose
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A Conjugate Transpose is a [[]] that ...
- See: Moore–Penrose Pseudoinverse, Mathematics, Matrix (Mathematics), Complex Number, Transpose, Complex Conjugate, Linear Algebra, Dagger (Typography), Quantum Mechanics.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/conjugate_transpose Retrieved:2015-3-1.
- In mathematics, the conjugate transpose or 'Hermitian 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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by : [math]\displaystyle{ (\boldsymbol{A}^*)_{ij} = \overline{\boldsymbol{A}_{ji}} }[/math] where the subscripts denote the i,j-th entry, for 1 ≤ i ≤ n and 1 ≤ j ≤ m, and the overbar denotes a scalar complex conjugate. (The complex conjugate of [math]\displaystyle{ a + bi }[/math] , where a and b are reals, is [math]\displaystyle{ a - bi }[/math] .)
This definition can also be written as : [math]\displaystyle{ \boldsymbol{A}^* = (\overline{\boldsymbol{A}})^\mathrm{T} = \overline{\boldsymbol{A}^\mathrm{T}} }[/math] where [math]\displaystyle{ \boldsymbol{A}^\mathrm{T} \,\! }[/math] denotes the transpose and [math]\displaystyle{ \overline{\boldsymbol{A}} \,\! }[/math] denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:
- [math]\displaystyle{ \boldsymbol{A}^* \,\! }[/math] or [math]\displaystyle{ \boldsymbol{A}^\mathrm{H} \,\! }[/math] , commonly used in linear algebra.
- [math]\displaystyle{ \boldsymbol{A}^\dagger \,\! }[/math] (sometimes pronounced as "A dagger"), universally used in quantum mechanics.
- [math]\displaystyle{ \boldsymbol{A}^+ \,\! }[/math] , although this symbol is more commonly used for the Moore–Penrose pseudoinverse.
- In some contexts, [math]\displaystyle{ \boldsymbol{A}^* \,\! }[/math] denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by [math]\displaystyle{ \boldsymbol{A}^{*\mathrm{T}} \,\! }[/math] or [math]\displaystyle{ \boldsymbol{A}^{\mathrm{T}*} \,\! }[/math] .
- In mathematics, the conjugate transpose or 'Hermitian 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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by : [math]\displaystyle{ (\boldsymbol{A}^*)_{ij} = \overline{\boldsymbol{A}_{ji}} }[/math] where the subscripts denote the i,j-th entry, for 1 ≤ i ≤ n and 1 ≤ j ≤ m, and the overbar denotes a scalar complex conjugate. (The complex conjugate of [math]\displaystyle{ a + bi }[/math] , where a and b are reals, is [math]\displaystyle{ a - bi }[/math] .)