matrices, over , and the Hilbert space of all complex matrices, . Eigenvectors and eigenvalues Each of the (Hermitian) Pauli matrices has two eigenvalues, and . The corresponding normalized eigenvectors are: Pauli vector The Pauli vector is defined by where , , and are an equivalent notation for the more familiar , , and ; the subscripted notation is more compact than the old form. The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows, using Einstein's summation convention. More formally, this defines a map from to the vector space of traceless Hermitian matrices. This map encodes structures of as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory. Another way to view the Pauli vector is as a Hermitian traceless matrix-valued dual vector, that is, an element of which maps . Completeness relation Each component of can be recovered from the matrix (see completeness relation below) This constitutes an inverse to the map , making it manifest that the map is a bijection. Determinant The norm is given by the determinant (up to a minus sign) Then considering the conjugation action of an matrix on this space of matrices, , we find , and that is Hermitian and traceless. It then makes sense to define where has the same norm as , and therefore interpret as a rotation of 3-dimensional space. In fact, it turns out that the special restriction on implies that the rotation is orientation preserving. This allows the definition of a map given by , where . This map is the concrete realization of the double cover of by , and therefore shows that . The components of can be recovered using the tracing process above: Cross-product The cross-product is given by the matrix commutator (up to a factor of ) In fact, the existence of a norm follows from the fact that is a Lie algebra: see Killing form. This cross-product can be used to prove the orientation-preserving property of the map above. Eigenvalues and eigenvectors The eigenvalues of are . This follows immediately from tracelessness and explicitly computing the determinant. More abstractly, without computing the determinant which requires explicit properties of the Pauli