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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
The Möbius–Kantor graph, the Cayley graph of the Pauli group with generators X, Y, and Z. In physics and mathematics, the Pauli group on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix and all of the Pauli matrices
The matrix is the 2×2 identity matrix and the matrices with =,, are the Pauli matrices. This decomposition simplifies the analysis of the system, especially in the time-independent case, where the values of α , β , γ {\displaystyle \alpha ,\beta ,\gamma } and δ {\displaystyle \delta } are constants.
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension . [2] [3] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum. The generalized Gell-Mann matrices are Hermitian and traceless by
where R T denotes the transpose of R and I is the 3 × 3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper and improper rotations. In addition to preserving length, proper rotations must also preserve orientation.
The center of SU(n) is isomorphic to the cyclic group /, and is composed of the diagonal matrices ζ I for ζ an n th root of unity and I the n × n identity matrix. Its outer automorphism group for n ≥ 3 is Z / 2 Z , {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} while the outer automorphism group of SU(2) is the trivial group .
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σ i, so that the Hermitian matrix is written as a Pauli vector. [2] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as ...