In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle \mathbf...

with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred...

that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the...

} 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...

matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose). The above definition of an adjoint operator...

mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*: A  normal ⟺ A ∗ A = A A ∗ . {\displaystyle A{\text{ normal}}\iff...

symbol ' denotes the conjugate transpose. The trivial modification is simply substituting the conjugate transpose for the real transpose everywhere. The advantages...

complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A {\displaystyle...

square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U U ∗ = I , {\displaystyle U^{*}U=UU^{*}=I...

of any degree Conjugate transpose, the complex conjugate of the transpose of a matrix Harmonic conjugate in complex analysis Conjugate (graph theory)...

vector transpose of x . {\displaystyle \mathbf {x} .} More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is...

Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers. The...

denotes the conjugate transpose of x {\displaystyle \mathbf {x} } (conjugate of the transpose, or equivalently transpose of the conjugate). Matrix multiplication...

^{m\times n}} ⁠, the transpose is denoted ⁠ A T {\displaystyle A^{\mathsf {T}}} ⁠ and the Hermitian transpose (also called conjugate transpose) is denoted ⁠...

different concept, the adjoint operator which for a matrix is the conjugate transpose. The product of a matrix with its adjugate gives a diagonal matrix...

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...

Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often...

\mathbf {U} ^{*}} denotes the conjugate transpose and U † {\displaystyle \mathbf {U} ^{\dagger }} denotes the conjugate transpose. They diagonalize using unitary...

unitary matrix, and V ∗ {\displaystyle \mathbf {V} ^{*}} is the conjugate transpose of ⁠ V {\displaystyle \mathbf {V} } ⁠. Such decomposition always...

} For complex vectors, it is often useful to take the conjugate transpose of v , {\displaystyle \mathbf {v} ,} denoted v † {\displaystyle \mathbf...

the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H: a ⋅ b = b H a . {\displaystyle \mathbf...

next to a ket implies matrix multiplication. The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice versa:...

reflector H=I-VTVH. "*larzb" applies a block reflector or its transpose/conjugate transpose as returned by "*tzrzf" to a general matrix. "*larzt" forms...

numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints...

{\displaystyle AA^{*}=A^{*}A} , where A ∗ {\displaystyle A^{*}} is a conjugate transpose) can be eigendecomposed. For a normal matrix A (and only for a normal...

a row vector that is the conjugate transpose to a column vector v {\displaystyle v} ). In quantum mechanics, the conjugate to a ket vector  | ψ ⟩ {\displaystyle...

they are represented, and some of their properties. Controlled or conjugate transpose (adjoint) versions of some of these gates may not be listed. The...

defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been...

reduces to that of being symmetric, and the conjugate transpose x ∗ {\displaystyle x^{*}} to the usual transpose x ′ {\displaystyle x'} . Note that R ( M...

positive-semidefinite matrix P can be written as N*N, where N* denotes the conjugate transpose of N (see Decomposition of a semidefinite matrix). Then det ( P )...