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  2. Conjugate transpose - Wikipedia

    en.wikipedia.org/wiki/Conjugate_transpose

    The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: [ 3 ]

  3. Hermitian matrix - Wikipedia

    en.wikipedia.org/wiki/Hermitian_matrix

    The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. [4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs. [5]

  4. Cholesky decomposition - Wikipedia

    en.wikipedia.org/wiki/Cholesky_decomposition

    In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.

  5. Transpose - Wikipedia

    en.wikipedia.org/wiki/Transpose

    A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if = ¯.

  6. Complex Hadamard matrix - Wikipedia

    en.wikipedia.org/wiki/Complex_Hadamard_matrix

    A complex Hadamard matrix is any complex ... where † denotes the Hermitian transpose of and is the identity matrix. The concept is a ...

  7. Unitary matrix - Wikipedia

    en.wikipedia.org/wiki/Unitary_matrix

    In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (⁠ † ⁠), so the equation above is written

  8. Conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Conjugate_gradient_method

    Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics , the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations , namely those whose matrix is positive-semidefinite .

  9. Jacobi method for complex Hermitian matrices - Wikipedia

    en.wikipedia.org/wiki/Jacobi_Method_for_Complex...

    A Hermitian matrix, H is defined by the conjugate transpose symmetry property: † = , =, By definition, the complex conjugate of a complex unitary rotation matrix, R is its inverse and also a complex unitary rotation matrix: