matrices, this follows from , since this can be factorised into . A standard result in linear algebra (a linear map which satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies is diagonal with possible eigenvalues . The tracelessness of means it has exactly one of each eigenvalue. Its normalized eigenvectors are Pauli 4-vector The Pauli 4-vector, used in spinor theory, is written with components This defines a map from to the vector space of Hermitian matrices, which also encodes the Minkowski metric (with mostly minus convention) in its determinant: This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector and allow raising and lowering using the Minkowski metric tensor. The relation can then be written Similarly to the Pauli 3-vector case, we can find a matrix group which acts as isometries on ; in this case the matrix group is , and this shows Similarly to above, this can be explicitly realized for with components In fact, the determinant property follows abstractly from trace properties of the . For matrices, the following identity holds: That is, the 'cross-terms' can be written as traces. When are chosen to be different , the cross-terms vanish. It then follows, now showing summation explicitly, Since the matrices are , this is equal to (Anti-)Commutation relations The Pauli matrices obey the following commutation relations: where the structure constant is the Levi-Civita symbol and Einstein summation notation is used. These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra They also satisfy the anticommutation relations: where is defined as , and is the Kronecker delta. denotes the identity matrix. These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for , denoted The usual construction of generators of using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors. A few explicit commutators and anti-commutators are given below as examples: Relation to dot and cross product Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives so that, Contracting each side of the equation with components of two -vectors and (which commute with the Pauli matrices, i.e., for each matrix and vector component (and likewise with ) yields Finally, translating the index notation for the dot product and cross