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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 eigenvectors may be found by the usual methods of linear algebra, but a convenient trick is to note that a Pauli spin matrix is an involutory matrix, that is, the square of the above matrix is the identity matrix. Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± S u. That is,
The traditional Pauli matrices are the matrix representation of the () Lie algebra generators , , and in the 2-dimensional irreducible representation of SU(2), corresponding to a spin-1/2 particle. These generate the Lie group SU(2) .
Suppose there is a spin 1/2 particle in a state = [].To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the adjoint of the eigenspinor matrix representing spin up, and square the result.
When spinors are used to describe the quantum states, the three spin operators (S x, S y, S z,) can be described by 2 × 2 matrices called the Pauli matrices whose eigenvalues are ± ħ / 2 . For example, the spin projection operator S z affects a measurement of the spin in the z direction.
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.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrödinger and Werner Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a two-component spinor wave-function. Pauli's theory of spin was non-relativistic.