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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.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
In the QR algorithm for a Hermitian matrix (or any normal matrix), the orthonormal eigenvectors are obtained as a product of the Q matrices from the steps in the algorithm. [11] For more general matrices, the QR algorithm yields the Schur decomposition first, from which the eigenvectors can be obtained by a backsubstitution procedure. [ 13 ] )
This method of generalizing the Pauli matrices refers to a generalization from a single 2-level system to multiple such systems. In particular, the generalized Pauli matrices for a group of qubits is just the set of matrices generated by all possible products of Pauli matrices on any of the qubits. [1]
POVM elements can be represented likewise, though the trace of a POVM element is not fixed to equal 1. The Pauli matrices are traceless and orthogonal to one another with respect to the Hilbert–Schmidt inner product, and so the coordinates (,,) of the state are the expectation values of the three von Neumann measurements defined by the Pauli ...
Using the cross product as a Lie bracket, the algebra of 3-dimensional real vectors is a Lie algebra isomorphic to the Lie algebras of SU(2) and SO(3). The structure constants are f a b c = ϵ a b c {\displaystyle f^{abc}=\epsilon ^{abc}} , where ϵ a b c {\displaystyle \epsilon ^{abc}} is the antisymmetric Levi-Civita symbol .
The Lie algebra of SO(3) is denoted by () and consists of all skew-symmetric 3 × 3 matrices. [7] This may be seen by differentiating the orthogonality condition , A T A = I , A ∈ SO(3) . [ nb 2 ] The Lie bracket of two elements of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} is, as for the Lie algebra of every matrix group, given by the ...
The Clifford algebra Cl 3,0 (R) has a faithful representation, generated by Pauli matrices, on the spin representation C 2; further, Cl 3,0 (R) is isomorphic to the even subalgebra Cl [0] 3,1 (R) of the Clifford algebra Cl 3,1 (R). APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.