konjugattransponering

Konjugattransponering, commonly denoted A*, is the conjugate transpose (also called the Hermitian transpose) of a complex matrix A. If A has entries a_ij, then A* is the transpose of A with each entry replaced by its complex conjugate. For real matrices, A* coincides with A^T.

Computation and notation: to form A*, first take the transpose, swapping rows and columns, and then take the complex conjugate of every element. The transpose is often denoted A^T, the Hermitian transpose A^H. Some texts use both A* and A^H to denote the same operation.

Properties: (A*)* = A. (AB)* = B* A*. (A+B)* = A* + B*. For any scalar c, (cA)* = c̄ A*. If A is square and invertible, (A−1)* = (A*)−1. A matrix is unitary if A* A = I = A A*. It is Hermitian if A* = A, and skew-Hermitian if A* = −A.

Spectral and inner-product implications: Hermitian matrices have real eigenvalues and orthogonal eigenvectors; unitary matrices preserve norms. The product A* A is Hermitian and positive semidefinite. In inner products, ⟨Ax, y⟩ = ⟨x, A* y⟩, illustrating A* as the adjoint operator with respect to the standard inner product.

Applications and example: Konjugattransponering is fundamental in solving linear systems via normal equations, in least squares, and in quantum mechanics as the adjoint operator. Example: A = [[1+i, 2], [3, −i]]; then A* = [[1−i, 3], [2, i]].