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Adjugate matrix. In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj (A). [1][2] It is also occasionally known as adjunct matrix, [3][4] or "adjoint", [5] though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate ...
Conjugate transpose. In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
t. e. In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n -by- n invertible matrices, then the adjoint representation is the group ...
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 ...
Its adjoint is † (), and the map † is complex linear in H. Thus H {\displaystyle H} embeds as a complex vector subspace of its own CCR algebra. In a representation of this algebra, the element a ( f ) {\displaystyle a(f)} will be realized as an annihilation operator, and a † ( f ) {\displaystyle a^{\dagger }(f)} as a creation operator.
Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem.Then the Lie algebra of G may be computed as [2] [3] = {(;)}. For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)
In mechanics and geometry, the 3D rotation group, often denoted SO (3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. [1] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation ...
Definition. An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism. ad X: g → g, ad X (Y) = [X, Y] is nilpotent, that is, (ad X) n = 0 for large enough n. Equivalently, X is nilpotent if its characteristic polynomial pad X (t) is equal to tdim g. A semisimple Lie group or algebraic group G acts on its Lie ...