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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]
The adjoint may also be called the Hermitian conjugate or simply the Hermitian [1] after Charles Hermite. It is often denoted by A † in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics.
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, (), which is also sometimes called adjoint. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } with real entries reduces to the transpose of A {\displaystyle \mathbf {A} } , as the conjugate of a real number is the number itself.
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
The adjoint M * of a complex matrix M is the transpose of the conjugate of M: M * = M T. A square matrix A is called normal if it commutes with its adjoint: A * A = AA *. It is called Hermitian if it is equal to its adjoint: A * = A. All Hermitian matrices are normal.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors however, use the term transpose to refer to the adjoint as defined here. The adjoint allows us to consider whether g : Y → X is equal to u −1 : Y → X.
In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, ... and Hermitian adjoint (Hermitian and skew-Hermitian matrices). ...