Conjugate transpose

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

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 ${\displaystyle a+bi}$, where a and b are reals, is ${\displaystyle a-bi}$.)

This definition can also be written as

where ${\displaystyle {\boldsymbol {A}}^{\mathrm {T} }}$ denotes the transpose and ${\displaystyle {\overline {\boldsymbol {A}}}}$ denotes the matrix with complex conjugated entries.

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